From cb01421262cc66948797e49807adb562797208fa Mon Sep 17 00:00:00 2001 From: Didnelpsun <2675350965@qq.com> Date: Wed, 2 Jun 2021 23:37:57 +0800 Subject: [PATCH] =?UTF-8?q?=E6=9B=B4=E6=96=B0=E4=B8=8D=E5=AE=9A=E7=A7=AF?= =?UTF-8?q?=E5=88=86=E4=B9=A0=E9=A2=98?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .../function-and-limit.pdf | Bin 0 -> 188359 bytes .../function-and-limit.tex} | 1437 +++++++++-------- advanced-math/exercise/1-limit/limit.pdf | Bin 177441 -> 0 bytes ...efinite-integral-and-definite-integral.pdf | Bin 0 -> 65958 bytes ...efinite-integral-and-definite-integral.tex | 147 ++ ...efinite-integral-and-definite-integral.pdf | Bin 166255 -> 166090 bytes ...efinite-integral-and-definite-integral.tex | 18 +- 7 files changed, 914 insertions(+), 688 deletions(-) create mode 100644 advanced-math/exercise/1-function-and-limit/function-and-limit.pdf rename advanced-math/exercise/{1-limit/limit.tex => 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advanced-math/exercise/1-function-and-limit/function-and-limit.tex index b04a9bc..95be339 100644 --- a/advanced-math/exercise/1-limit/limit.tex +++ b/advanced-math/exercise/1-function-and-limit/function-and-limit.tex @@ -1,679 +1,758 @@ -\documentclass[UTF8, 12pt]{ctexart} -% UTF8编码,ctexart现实中文 -\usepackage{color} -% 使用颜色 -\definecolor{orange}{RGB}{255,127,0} -\usepackage{geometry} -\setcounter{tocdepth}{4} -\setcounter{secnumdepth}{4} -% 设置四级目录与标题 -\geometry{papersize={21cm,29.7cm}} -% 默认大小为A4 -\geometry{left=3.18cm,right=3.18cm,top=2.54cm,bottom=2.54cm} -% 默认页边距为1英尺与1.25英尺 -\usepackage{indentfirst} -\setlength{\parindent}{2.45em} -% 首行缩进2个中文字符 -\usepackage{setspace} -\renewcommand{\baselinestretch}{1.5} -% 1.5倍行距 -\usepackage{amssymb} -% 因为所以 -\usepackage{amsmath} -% 数学公式 -\usepackage{pifont} -% 圆圈序号 -\usepackage{mathtools} -% 有字的长箭头 -\usepackage[colorlinks,linkcolor=black,urlcolor=blue]{hyperref} -% 超链接 -\author{Didnelpsun} -\title{极限} -\date{} -\begin{document} -\maketitle -\thispagestyle{empty} -\tableofcontents -\thispagestyle{empty} -\newpage -\pagestyle{plain} -\setcounter{page}{1} - -\section{极限类型} - -七种:$\dfrac{0}{0},\dfrac{\infty}{\infty},0\cdot\infty,\infty-\infty,\infty^0,0^0,1^\infty$。 - -\medskip - -\ding{172}其中$\dfrac{0}{0}$为洛必达法则的基本型。$\dfrac{\infty}{\infty}$可以类比$\dfrac{0}{0}$的处理方式。$0\cdot\infty$可以转为$\dfrac{0}{\dfrac{1}{\infty}}=\dfrac{0}{0}=\dfrac{\infty}{\dfrac{1}{0}}=\dfrac{\infty}{\infty}$。设置分母有原则,简单因式才下放(简单:幂函数,e为底的指数函数)。 - -\medskip - -\ding{173}$\infty-\infty$可以提取公因式或通分,即和差化积。 - -\medskip - -\ding{174}$\infty^0,0^0,1^\infty$,就是幂指函数。 - -\medskip - -$ -u^v=e^{v\ln u}=\left\{ -\begin{array}{lcl} - \infty^0 & \rightarrow & e^{0\cdot+\infty} \\ - 0^0 & \rightarrow & e^{0\cdot-\infty} \\ - 1^\infty & \rightarrow & e^{\infty\cdot 0} \\ -\end{array} \right. -$ - -\medskip - -$\therefore \lim u^v=e^{\lim v\cdot\ln u}=e^{\lim v(u-1)}$ - -综上,无论什么样的四则形式,都必须最后转换为商的形式。 - -\section{常用化简运算} - -\subsection{对数法则} - -\textbf{例题:}求$\lim\limits_{x\to 0}\dfrac{(e^{x^2}-1)(\sqrt{1+x}-\sqrt{1-x})}{[\ln(1-x)+\ln(1+x)]\sin\dfrac{x}{x+1}}$。 - -注意在积或商的时候不能把对应的部分替换为0,如分母部分的$[\ln(1-x)+\ln(1+x)]$就无法使用$\ln(1+x)\sim x$替换为$-x+x$,这样底就是0了,无法求得最后的极限。 - -这时可以尝试变形,如对数函数相加等于对数函数内部式子相乘:$\ln(1-x)+\ln(1+x)=\ln(1-x^2)\sim-x^2$。 - -\subsection{指数法则} - -一般需要与洛必达法则配合使用。\medskip - -\textbf{例题:}求$\lim\limits_{x\to 0}\left(\dfrac{a^x+b^x+c^x}{3}\right)^{\frac{1}{x}}(a>0,b>0,c>0)$。\medskip - -$=e^{\lim\limits_{x\to 0}\frac{\ln\left(\frac{a^x+b^x+c^x}{3}\right)}{x}}=e^{\lim\limits_{x\to 0}\frac{\ln(a^x+b^x+c^x)-\ln 3}{x}}=e^{\lim\limits_{x\to 0}\frac{a^x\ln a+b^x\ln b+c^x\ln c}{a^x+b^x+c^x}}$(洛必达法则) - -$=e^{\lim\limits_{x\to 0}\frac{\ln a+\ln b+\ln c}{1+1+1}}=e^{\lim\limits_{x\to 0}\frac{\ln(abc)}{3}}=\sqrt[3]{abc}$。 - -\textbf{例题:}求$\lim\limits_{n\to\infty}n\left[\left(1+\dfrac{1}{n}\right)^{\frac{n}{2}}-\sqrt{e}\right]$。 - -首先对于幂指函数需要取指数,所以$\left(1+\dfrac{1}{n}\right)^{\frac{n}{2}}=e^{\frac{n}{2}\ln(1+\frac{1}{n})}$。\medskip - -而后面的多一个$\sqrt{e}$导致整个式子变为一个复杂的式子,而与$e^x$相关的是$e^x-1\sim x$。 - -所以$e^{\frac{n}{2}\ln(1+\frac{1}{n})}-\sqrt{e}=e^{\frac{1}{2}}\cdot\left(e^{\frac{n}{2}\ln(1+\frac{1}{n})-\frac{1}{2}}-1\right)=e^{\frac{1}{2}}\cdot\left[\dfrac{n}{2}\ln(1+\dfrac{1}{n})-\dfrac{1}{2}\right]$。 - -综上: - -$\lim\limits_{n\to\infty}n\left[\left(1+\dfrac{1}{n}\right)^{\frac{n}{2}}-\sqrt{e}\right]=\lim\limits_{n\to\infty}n\left(e^{\frac{n}{2}\ln(1+\frac{1}{n})}-\sqrt{e}\right)$ \medskip - -$=\lim\limits_{n\to\infty}n\left[e^{\frac{1}{2}}\cdot\left(e^{\frac{n}{2}\ln(1+\frac{1}{n})-\frac{1}{2}}-1\right)\right]=\dfrac{e^{\frac{1}{2}}}{2}\lim\limits_{n\to\infty}n^2\left[\ln\left(1+\frac{1}{n}\right)-\dfrac{1}{n}\right]$ - -$=\dfrac{e^{\frac{1}{2}}}{2}\lim\limits_{n\to\infty}\dfrac{\dfrac{1}{n}-\dfrac{1}{2n^2}-\dfrac{1}{n}}{\dfrac{1}{n^2}}=\dfrac{e^{\frac{1}{2}}}{2}\cdot\left(-\dfrac{1}{2}\right)=-\dfrac{\sqrt{e}}{4}$ - -\subsection{三角函数关系式} - -\textbf{例题:}求极限$\lim\limits_{x\to 0}\left(\dfrac{1}{\sin^2x}-\dfrac{\cos^2x}{x^2}\right)$。\medskip - -$\lim\limits_{x\to 0}\left(\dfrac{1}{\sin^2x}-\dfrac{\cos^2x}{x^2}\right)=\lim\limits_{x\to 0}\dfrac{x^2-\sin^2x\cos^2x}{\sin^2x\cdot x^2} (\sin x\sim x)$ \medskip - -$=\lim\limits_{x\to 0}\dfrac{x^2-\sin^2x\cos^2x}{x^4} (\sin x\cos x\sim\dfrac{1}{2}\sin 2x)=\lim\limits_{x\to 0}\dfrac{x^2-\dfrac{1}{4}\sin^22x}{x^4}$ \medskip - -$=\lim\limits_{x\to 0}\dfrac{2x-\dfrac{1}{4}\cdot 2\sin 2x\cdot\cos 2x\cdot 2}{4x^3} (\sin x\cos x\sim\dfrac{1}{2}\sin 2x)=\lim\limits_{x\to 0}\dfrac{2x-\dfrac{1}{2}\sin 4x}{4x^3}$ \medskip - -$=\lim\limits_{x\to 0}\dfrac{2-\dfrac{1}{2}\cos 4x\cdot 4}{12x^2}=\dfrac{1}{6}\lim\limits_{x\to 0}\dfrac{1-\cos 4x}{x^2} (1-\cos x\sim \dfrac{1}{2}x^2)=\dfrac{4}{3}$ - -\subsection{提取常数因子} - -提取常数因子就是提取出能转换为常数的整个极限式子的因子。这个因子必然在自变量的趋向时会变为非0的常数,那么这个式子就可以作为常数提出。 - -\subsection{提取公因子} - -当作为商的极限式子上下都具有公因子时可以提取公因子然后相除,从而让未知数集中在分子或分母上。 - -\textbf{例题:}求$\lim\limits_{x\to 4}\dfrac{x^2-6x+8}{x^5-5x+4}$。 - -需要先提取公因子: - -$=\lim\limits_{x\to 4}\dfrac{(x-2)(x-4)}{(x-1)(x-4)}=\lim\limits_{x\to 4}\dfrac{x-2}{x-1}=\dfrac{2}{3}$。 - -(当然可以使用洛必达法则得到极限为$\lim\limits_{x\to 4}\dfrac{2x-6}{2x-5}=\lim\limits_{x\to 4}\dfrac{8-6}{8-5}$) - -\textcolor{orange}{注意:}提取公因子的时候应该注意开平方等情况下符号的问题。如果极限涉及倒正负两边则必须都讨论。 - -当趋向为负且式子中含有根号的时候最好提取负因子,从而让趋向变为正。\medskip - -\textbf{例题:}求$\lim\limits_{x\to-\infty}\left[\sqrt{4x^2+x}\ln\left(2+\dfrac{1}{x}\right)+2\ln 2x\right]$。\medskip - -题目的形式为$\infty-\infty$,所以必须使用后面的倒代换转换为商的形式。\medskip - -$=\lim\limits_{x\to-\infty}-x\left[\sqrt{4+\dfrac{1}{x}}\ln\left(2+\dfrac{1}{x}\right)-2\ln 2\right]$。 \medskip - -这里就需要注意到因为$\sqrt{4x^2+x}$的限制导致这个式子必然为正数,而$x\to-\infty$代表自变量为负数,所以提出来的$x$必然是负数,而原式是正数,所以就需要添加一个负号,而后面的$2\ln 2x$则没有要求,所以直接变成$-2\ln 2$就可以了。 - -将$x$下翻变成分母为$\dfrac{1}{x}$,并令$t=\dfrac{1}{x}$。\medskip - -$=\lim\limits_{t\to 0^-}\dfrac{\sqrt{t+4}\ln\left(2+\dfrac{1}{x}\right)-2\ln 2}{-t}$。\medskip - -幂次不高可以尝试洛必达:\medskip - -$=\lim\limits_{t\to 0^-}\left(\dfrac{1}{2}\cdot\dfrac{\ln(2+t)}{\sqrt{t+4}}+\dfrac{\sqrt{t+4}}{2+t}\right)=-\left(\dfrac{1}{2}\cdot\dfrac{\ln 2}{2}+\dfrac{2}{2}\right)=-\dfrac{\ln 2}{4}-1$。 - -\subsection{幂指函数} - -当出现$f(x)^{g(x)}$的类似幂函数与指数函数类型的式子,需要使用$u^v=e^{v\ln u}$。 - -\textbf{例题:}求极限$\lim\limits_{x\to+\infty}(x+\sqrt{1+x^2})^{\frac{1}{x}}$。 - -$\lim\limits_{x\to+\infty}(x+\sqrt{1+x^2})^{\frac{1}{x}}=e^{\lim\limits_{x\to+\infty}\frac{\ln(x+\sqrt{1+x^2})}{x}} \left(\ln(x+\sqrt{1+x^2})'=\dfrac{1}{\sqrt{1+x^2}}\right)$\medskip - -$=e^{\lim\limits_{x\to+\infty}\frac{1}{\sqrt{1+x^2}}}=e^0=1$ - -\subsection{有理化} - -当遇到带有根号的式子可以使用等价无穷小,但是只针对形似$(1+x)a-1\sim ax$的式子,而针对$x^a\pm x^b$的式子则无法替换,必须使用有理化来将单个式子变为商的形式。 - -如$\sqrt{a}\pm\sqrt{b}=\dfrac{a+b}{\sqrt{a}\mp\sqrt{b}}$。\medskip - -\textbf{例题:}求极限$\lim\limits_{x\to-\infty}x(\sqrt{x^2+100}+x)$。 - -首先定性分析:$\lim\limits_{x\to-\infty}x\cdot(\sqrt{x^2+100}+x)$。 - -在$x\to-\infty$趋向时,$x$就趋向无穷大。 - -而$\sqrt{x^2+100}$为一次,所以$\sqrt{x^2+100}+x$趋向0。 - -又$\sqrt{x^2+100}$在$x\to-\infty$时本质为根号差,所以有理化: - -$\lim\limits_{x\to-\infty}x(\sqrt{x^2+100}+x)=\lim\limits_{x\to-\infty}x\dfrac{x^2+100-x^2}{\sqrt{x^2+100}-x}=\lim\limits_{x\to-\infty}\dfrac{100x}{\sqrt{x^2+100}-x}$\medskip - -$\xRightarrow{\text{令}x=-t}\lim\limits_{t\to+\infty}\dfrac{-100t}{\sqrt{t^2+100}+t}=\lim\limits_{t\to+\infty}\dfrac{-100}{\sqrt{1+\dfrac{100}{t^2}}+1}=-50$\medskip - -\textbf{例题:}求$\lim\limits_{x\to 0}\dfrac{\sqrt{1+\tan x}-\sqrt{1+\sin x}}{x\sqrt{1+\sin^2x}-x}$。 - -$=\lim\limits_{x\to 0}\sqrt{1+\tan x}-\sqrt{1+\sin x}\cdot\lim\limits_{x\to 0}\dfrac{1}{x\sqrt{1+\sin^2x}-x}$\medskip - -$=\lim\limits_{x\to 0}\dfrac{\tan x-\sin x}{\sqrt{1+\tan x}+\sqrt{1+\sin x}}\cdot\lim\limits_{x\to 0}\dfrac{x\sqrt{1+\sin^2x}+x}{x^2(1+\sin^2x)-x^2}$\medskip - -$=\lim\limits_{x\to 0}\dfrac{\tan x-\sin x}{\sqrt{1+\tan x}+\sqrt{1+\sin x}}\cdot\lim\limits_{x\to 0}\dfrac{\sqrt{1+\sin^2x}+1}{x\sin^2x}$\medskip - -$=\lim\limits_{x\to 0}\dfrac{\tan x-\sin x}{2}\cdot\lim\limits_{x\to 0}\dfrac{2}{x\sin^2x}$ - -$=\lim\limits_{x\to 0}\dfrac{\tan x-\sin x}{x\sin^2x}=\lim\limits_{x\to 0}\dfrac{1-\cos x}{x\cos x\sin x}=\lim\limits_{x\to 0}\dfrac{\dfrac{1}{2}x^2}{x^2}=\dfrac{1}{2}$。 - -\subsection{换元法} - -换元法本身没什么技巧性,主要是更方便计算。最重要的是获取到共有的最大因子进行替换。 - -\textbf{例题:}求极限$\lim\limits_{x\to 1^-}\ln x\ln(1-x)$。 - -当$x\to 1^-$时,$\ln x$趋向0,$\ln(1-x)$趋向$-\infty$。 - -又$x\to 0$,$\ln(1+x)\sim x$,所以$x\to 1$,$\ln x\sim x-1$: - -$\lim\limits_{x\to 1^-}\ln x\ln(1-x)=\lim\limits_{x\to 1^-}(x-1)\ln(1-x)\xRightarrow{令t=1-x} =-\lim\limits_{t\to 0^+}t\ln t$ - -$=-\lim\limits_{t\to 0^+}\dfrac{\ln t}{\dfrac{1}{t}}=-\lim\limits_{t\to 0^+}\dfrac{\dfrac{1}{t}}{-\dfrac{1}{t^2}}=\lim\limits_{t\to 0^+}t=0$ - -\subsection{倒代换} - -\subsubsection{含有分式} - -当极限式子中含有分式中一般都需要用其倒数,把分式换成整式方便计算。 - -\textbf{例题:}求极限$\lim\limits_{x\to 0}\dfrac{e^{-\frac{1}{x^2}}}{x^{100}}$ \medskip - -$\lim\limits_{x\to 0}\dfrac{e^{-\frac{1}{x^2}}}{x^{100}}=\lim\limits_{x\to 0}\dfrac{e^{-\frac{1}{x^2}}\cdot 2x^{-3}}{100x^{99}}=\lim\limits_{x\to 0}\dfrac{1}{50}\lim\limits_{x\to 0}\dfrac{e^{-\frac{1}{x^2}}}{x^{102}}$ - -\medskip - -使用洛必达法则下更复杂,因为分子的幂次为负数,导致求导后幂次绝对值越来越大,不容易计算。 - -使用倒代换再洛必达降低幂次,令$t=\dfrac{1}{x^2}$ - -$\lim\limits_{x\to 0}\dfrac{e^{-\frac{1}{x^2}}}{x^{100}}=\lim\limits_{t\to+\infty}\dfrac{e^{-t}}{t^{-50}}=\lim\limits_{t\to+\infty}\dfrac{t^{50}}{e^t}=\lim\limits_{t\to+\infty}\dfrac{50t^{49}}{e^t}$ - -$=\cdots$ - -$=\lim\limits_{t\to+\infty}\dfrac{50!}{e^t}=0$ - -\textbf{例题:}求极限$\lim\limits_{x\to+\infty}[x^2(e^{\frac{1}{x}}-1)-x]$。 - -该式子含有分数,所以尝试使用倒数代换:\medskip - -$\lim\limits_{x\to+\infty}[x^2(e^{\frac{1}{x}}-1)-x]\xRightarrow{\text{令}x=\frac{1}{t}}\lim\limits_{t\to 0^+}\left(\dfrac{e^t-1}{t^2}-\dfrac{1}{t}\right)=\lim\limits_{t\to 0^+}\dfrac{e^t-1-t}{t^2}$ - -$\xRightarrow{\text{泰勒展开}e^t}\lim\limits_{t\to 0^+}\dfrac{\dfrac{1}{2}t^2}{t^2}=\dfrac{1}{2}$ - -\subsubsection{\texorpdfstring{$\infty-\infty$}\ 型} - -\subsection{拆项} - -拆项需要根据式子形式进行,所以很难找到普遍规律。 - -\textbf{例题:}求$\lim\limits_{n\to\infty}\dfrac{(n+1)(n+2)(n+3)\cdots(n+6)}{6n^6}$。 - -需要将分子和分母都拆为6项: - -$=\dfrac{1}{6}\lim\limits_{n\to\infty}\dfrac{n+1}{n}\times\dfrac{n+2}{n}\times\cdots\dfrac{n+6}{n}=\dfrac{1}{6}\lim\limits_{n\to\infty}(1+\dfrac{1}{n})(1+\dfrac{2}{n})\cdots(1+\dfrac{6}{n})=\dfrac{1}{6}$。 - -当极限式子中出现不知道项数的$n$时,一般需要使用拆项,把项重新组合。一般的组合是根据等价无穷小。 - -而对于复杂的具有同一结构的式子也可以考虑拆项。 - -\textbf{例题:}求极限$\lim\limits_{x\to 0}\left(\dfrac{e^x+e^{2x}+\cdots+e^{nx}}{n}\right)^{\frac{e}{x}}$。($n\in N^+$) - -这里可以使用等价无穷小$e^x-1\sim x$。 - -$\lim\limits_{x\to 0}\left(\dfrac{e^x+e^{2x}+\cdots+e^{nx}}{n}\right)^{\frac{e}{x}}=e^{\lim\limits_{x\to 0}\frac{e}{x}\ln\left(\frac{e^x+e^{2x}+\cdots+e^{nx}}{n}\right)}$ - -$=e^{\lim\limits_{x\to 0}\frac{e}{x}\left(\frac{e^x+e^{2x}+\cdots+e^{nx}}{n}-1\right)}=e^{\lim\limits_{x\to 0}\frac{e}{x}\left(\frac{e^x+e^{2x}+\cdots+e^{nx}-n}{n}\right)}$ - -$=e^{\frac{e}{n}\lim\limits_{x\to 0}\left(\frac{e^x-1}{x}+\frac{e^{2x}-1}{x}+\cdots+\frac{e^{nx}-1}{x}\right)}=e^{\frac{e}{n}[1+2+\cdots+n]}=e^{\frac{e}{n}\cdot\frac{n(1+n)}{2}}=e^{\frac{e(1+n)}{2}}$ - -\textbf{例题:}求$\lim\limits_{x\to 0}\dfrac{1-\cos x\sqrt{\cos 2x}\sqrt[3]{\cos 3x}}{\ln\cos x}$\medskip - -可以使用$\cos x-1\sim\dfrac{x^2}{2}$。\medskip - -$\lim\limits_{x\to 0}\dfrac{1-\cos x\sqrt{\cos 2x}\sqrt[3]{\cos 3x}}{\ln\cos x}$ - -$=\lim\limits_{x\to 0}\dfrac{1-\cos x+\cos x-\cos x\sqrt{\cos 2x}\sqrt[3]{\cos 3x}}{-\dfrac{x^2}{2}}$ - -$=\lim\limits_{x\to 0}\dfrac{\dfrac{x^2}{2}}{-\dfrac{x^2}{2}}+\dfrac{\cos x(1-\sqrt{\cos 2x}\sqrt[3]{\cos 3x})}{-\dfrac{x^2}{2}}$ - -$=-1+\lim\limits_{x\to 0}\dfrac{(1-\sqrt{\cos 2x})+\sqrt{\cos 2x}-\sqrt{\cos 2x}\sqrt[3]{\cos 3x}}{-\dfrac{x^2}{2}}$ - -$=-1+\lim\limits_{x\to 0}\dfrac{-\dfrac{1}{2}(\cos 2x-1)+\sqrt{\cos 2x}(1-\sqrt[3]{\cos 3x})}{-\dfrac{x^2}{2}}$ - -$=-1+\lim\limits_{x\to 0}\dfrac{-\dfrac{1}{2}(-\dfrac{4x^2}{2})+\left(-\dfrac{1}{3}\right)\left(-\dfrac{9x^2}{2}\right)}{-\dfrac{x^2}{2}}=-6$ - -\section{基本计算方式} - -课本上极限计算可以使用的主要计算方式: - -\subsection{基础四则运算} - -只有式子的极限各自存在才能使用四则运算,使用的频率较少。 - -\subsection{重要极限} - -重要极限有两个,但是$\lim\limits_{x\to 0}\dfrac{\sin x}{x}=1$这个很少用,因为往往用等价无穷小替代了,而$\lim\limits_{x\to\infty}\left(1+\dfrac{1}{x}\right)^x=e$则用的较多,当出现分数幂的幂指函数时,不要先去取对数,而是使用重要极限看看能不能转换。\medskip - -\textbf{例题:}求$\lim\limits_{x\to\infty}\left(\dfrac{3+x}{6+x}\right)^{\frac{x-1}{2}}$。\medskip - -$=\lim\limits_{x\to\infty}\left(1-\dfrac{3}{6+3}\right)^{\frac{6+x}{-3}\cdot\frac{-3}{6+x}\cdot\frac{x-1}{2}}=\lim\limits_{x\to\infty}e^{\frac{-3}{6+x}\cdot\frac{x-1}{2}}=\lim\limits_{x\to\infty}e^{-\frac{3}{2}\cdot\frac{x-1}{x+6}}=e^{-\frac{3}{2}}$。\medskip - -\textbf{例题:}求$\lim\limits_{x\to\infty}\left(\dfrac{2x+3}{2x+1}\right)^{x+1}$。\medskip - -$=\lim\limits_{x\to\infty}\left(\dfrac{2x+3}{2x+1}\right)^x\cdot\dfrac{2x+3}{2x+1}=\lim\limits_{x\to\infty}\left(\dfrac{2x+3}{2x+1}\right)^x$\medskip - -$=\lim\limits_{x\to\infty}\left(\dfrac{1+\dfrac{3}{2x}}{1+\dfrac{1}{2x}}\right)^x=\lim\limits_{x\to\infty}\dfrac{\left(1+\dfrac{3}{2x}\right)^x}{\left(1+\dfrac{1}{2x}\right)^x}$ - -$=\lim\limits_{x\to\infty}\dfrac{\left[\left(1+\dfrac{3}{2x}\right)^{\frac{2x}{3}}\right]^{\frac{3}{2}}}{\left[\left(1+\dfrac{1}{2x}\right)^{2x}\right]^{\frac{1}{2}}}=\dfrac{e^{\frac{3}{2}}}{e^{\frac{1}{2}}}=e$。 - -\subsection{导数定义} - -极限转换以及连续性的时候会用到,但是使用的频率也较小。 - -\subsection{等价无穷小替换} - -当看到复杂的式子,且不论要求的极限值的趋向,而只要替换的式子是$\Delta\to 0$时的无穷小,就使用等价无穷小进行替换。 - -\textcolor{orange}{注意:}替换的必然是整个求极限的乘或除的因子,一般加减法与部分的因子不能进行等价无穷小替换。 - -对于无法直接得出变换式子的,可以对对应参数进行凑,以达到目标的可替换的等价无穷小。 - -\subsection{夹逼准则} - -夹逼准则可以用来证明不等式也可以用来计算极限。但是最重要的是找到能夹住目标式子的两个式子。\medskip - -\textbf{例题:}求极限$\lim\limits_{x\to 0}x\left[\dfrac{10}{x}\right]$,其中$[\cdot]$为取整符号。 - -取整函数公式:$x-1<[x]\leqslant x$,所以$\dfrac{10}{x}-1<\left[\dfrac{10}{x}\right]\leqslant\dfrac{10}{x}$。 - -当$x>0$时,$x\to 0^+$,两边都乘以10,$10-x0$时,$x\to 0^-$,同样也是夹逼准则得到极限为10。\medskip - -$\therefore \lim\limits_{x\to 0}x\left[\dfrac{10}{x}\right]=10$。 - -\subsection{拉格朗日中值定理} - -对于形如$f(a)-f(b)$的极限式子就可以使用拉格朗日中值定理,这个$f(x)$为任意的函数。使用拉格朗日中值定理最重要的还是找到这个$f(x)$。 - -\subsubsection{证明不等式} - -证明不等式最重要的还是找到$f(x)$,有时候不等式不存在$f(a)-f(b)$这种式子,就需要我们转换。 - -\paragraph{对数函数特性} \leavevmode \medskip - -对于对数函数,要记住其特定的性质:$\log_n(\dfrac{a}{b})=\log_na-\log_nb$。 - -\textbf{例题:}设$a>b>0$,证明:$\dfrac{a-b}{a}<\ln\dfrac{a}{b}<\dfrac{a-b}{b}$。 - -因为$\ln\dfrac{a}{b}=\ln a-\ln b$,所以令$f(x)=\ln x$。 - -所以根据拉格朗日中值定理:$\ln a-\ln b=f'(\xi)(a-b)$($\xi\in(b,a)$)。 - -又$f'(\xi)=\dfrac{1}{\xi}$,所以$\ln a-\ln b=\dfrac{a-b}{\xi}$。 - -又$\xi\in(b,a)$,所以$\dfrac{1}{\xi}\in(\dfrac{1}{a},\dfrac{1}{b})$。 - -所以$\dfrac{a-b}{a}<\dfrac{a-b}{\xi}<\dfrac{a-b}{b}$,从而$\dfrac{a-b}{a}<\ln\dfrac{a}{b}<\dfrac{a-b}{b}$,得证。 - -\paragraph{查找特定值} \leavevmode \medskip - -对于证明一种不等式,如果里面没有差式,也无法转换为差式,那么就可以考虑制造差式,对于$f(x)$一般选择更高阶的,$a$选择$x$,$b$要根据题目和不等式设置一个常数。 - -一般是0或1。可以先尝试1。 - -\textbf{例题:}当$x>1$时,证明$e^x>ex$。 - -题目中没有差式,所以需要选择一个函数作为基准函数,里面有一个指数函数和一个幂函数,所以选择$e^x$作为基准函数。 - -然后选择一个常数作为$b$值,可以先选一个1作为$b$值:$f(x)-f(1)=f'(\xi)(x-1)$。 - -从而$e^x-e=e^\xi(x-1)$,$\xi\in(1,x)$,所以$e^x-e>e(x-1)$,即$e^x>ex$,得证。 - -\subsubsection{极限计算} - -可以将极限式子中形如$f(a)-f(b)$的极限部分使用拉格朗日中值定理进行替换。 - -\textbf{例题:}求极限$\lim\limits_{n\to\infty}n^2\left(\arctan\dfrac{2}{n}-\arctan\dfrac{2}{n+1}\right)$。\medskip - -因为式子不算非常复杂,其实也可以通过洛必达法则来完成,但是求导会很复杂。而$\arctan x$可以认定为$f(x)$。 - -从而$\arctan\dfrac{2}{n}-\arctan\dfrac{2}{n+1}$为$f(\dfrac{2}{n})-f(\dfrac{2}{n+1})=f'(\xi)\left(\dfrac{2}{n}-\dfrac{2}{n+1}\right)$。 - -其中$\dfrac{2}{n+1}<\xi<\dfrac{2}{n}$,而当$n\to\infty$时,$f'(\xi)=\dfrac{1}{1+\xi^2}\to 1$。 - -$\therefore\arctan\dfrac{2}{n}-\arctan\dfrac{2}{n+1}\sim\dfrac{2}{n}-\dfrac{2}{n+1}=\dfrac{2}{n(n+1)}$。 - -$\therefore\lim\limits_{n\to\infty}n^2\left(\arctan\dfrac{2}{n}-\arctan\dfrac{2}{n+1}\right)=\lim\limits_{n\to\infty}n^2\cdot\dfrac{2}{n(n+1)}=2$。 - -\subsection{洛必达法则} - -洛必达法则的本质是降低商形式的极限式子的幂次。 - -洛必达在处理一般的极限式子比较好用,但是一旦式子比较复杂最好不要使用洛必达法则,最好是对求导后有规律或幂次较低的式子进行上下求导。 - -对于幂次高的式子必然使用洛必达法则。 - -洛必达法则必须使用在分式都趋向0或$\infty$时,如果不是这样的趋向则不能使用。如: - -\textbf{例题:}求$\lim\limits_{x\to 1}\dfrac{x^2-x+1}{(x-1)^2}$。 - -如果使用洛必达法则,则会得到结果为1,这是错误的,因为分子在$x\to 1$时结果为常数1。正确的计算方式: - -$=\lim\limits_{x\to 1}\dfrac{1}{(x-1)^2}=\infty$。 - -\subsection{泰勒公式} - -泰勒公式一般会使用趋向0的麦克劳林公式,且一般只作为极限计算的一个小部分,用来替代一个部分。 - -且一般只有麦克劳林公式表上的基本初等函数才会使用倒泰勒公式,复合函数最好不要使用。 - -\textbf{例题:}求极限$\lim\limits_{x\to 0}\dfrac{\arcsin x-\arctan x}{\sin x-\tan x}$。\medskip - -分析:该题目使用洛必达法则会比较麻烦且难以计算,所以先考虑是否能用泰勒展开: - -$x\to 0$,$\sin x=x-\dfrac{1}{6}x^3+o(x^3)$,$\tan x=x+\dfrac{1}{3}x^3+o(x^3)$,$\arcsin x=x+\dfrac{1}{6}x^3+o(x^3)$,$\arctan x=x-\dfrac{1}{3}x^3+o(x^3)$。 - -$\therefore \sin x-\tan x=-\dfrac{1}{2}x^3+o(x^3)$,$\arcsin x-\arctan x=\dfrac{1}{2}x^3+o(x^3)$ - -$\therefore \text{原式}=\dfrac{\dfrac{1}{x}x^3+o(x^3)}{-\dfrac{1}{2}x^3+o(x^3)}=-1$。 - -\section{极限计算形式} - -极限相关计算形式主要分为下面六种: - -\begin{enumerate} - \item 未定式:直接根据式子计算极限值。 - \item 极限转换:根据已知的极限值计算目标极限值。 - \item 求参数:已知式子的极限值,计算式子中未知的参数。 - \item 极限存在性:根据式子以及极限存在性计算极限或参数。 - \item 极限唯一性:式子包含参数,根据唯一性计算两侧极限并求出参数与极限值。 - \item 函数连续性:根据连续性与附加条件计算极限值或参数。 - \item 迭代式数列:根据数列迭代式计算极限值。 - \item 变限积分:根据变限积分计算极限值。 -\end{enumerate} - -\subsection{极限转换} - -\subsubsection{整体换元} - -最常用的方式就是将目标值作为一个部分,然后对已知的式子进行替换。 - -\textbf{例题:}已知$\lim\limits_{x\to 0}\dfrac{\ln(1-x)+xf(x)}{x^2}=0$,求$\lim\limits_{x\to 0}\dfrac{f(x)-1}{x}$。\medskip - -令目标$\dfrac{f(x)-1}{x}=t$,$\therefore f(x)=tx+1$。\medskip - -$\lim\limits_{x\to 0}\dfrac{\ln(1-x)+xf(x)}{x^2}=\lim\limits_{x\to 0}\dfrac{\ln(1-x)+tx^2+x}{x^2} (\text{泰勒展开})$\medskip - -$=\lim\limits_{x\to 0}\dfrac{-x-\dfrac{x^2}{2}+tx^2+x}{x^2}=\lim\limits_{x\to 0}\dfrac{\left(t-\dfrac{1}{2}\right)x^2}{x^2}=\lim\limits_{x\to 0}\left(t-\dfrac{1}{2}\right)=0$ - -$\therefore\lim\limits_{x\to 0}t=\lim\limits_{x\to 0}\dfrac{f(x)-1}{x}=\dfrac{1}{2}$。 - -\subsubsection{关系转换} - -\textbf{例题:}如果$\lim\limits_{x\to 0}\dfrac{x-\sin x+f(x)}{x^4}$存在,则$\lim\limits_{x\to 0}\dfrac{x^3}{f(x)}$为常数多少? - -由$\lim\limits_{x\to 0}\dfrac{x\sin x+f(x)}{x^4}=A$,而目标是$x^3$,所以需要变形: - -$\lim\limits_{x\to 0}\dfrac{x\sin x+f(x)}{x^4}=A$ - -$\lim\limits_{x\to 0}\dfrac{x\sin x+f(x)\cdot x}{x^4}=A\cdot\lim\limits_{x\to 0}x=0$ - -$\lim\limits_{x\to 0}\dfrac{x-\sin x}{x^3}+\lim\limits_{x\to 0}\dfrac{f(x)}{x^3}=0$ - -$\text{泰勒展开:}x-\sin x=\dfrac{1}{6}x^3$ - -$\lim\limits_{x\to 0}\dfrac{f(x)}{x^3}=-\dfrac{1}{6}$ - -$\lim\limits_{x\to 0}\dfrac{x^3}{f(x)}=-6$ - -\subsubsection{脱帽法} - -$\lim\limits_{x\to x_0}f(x)\Leftrightarrow f(x)=A+\alpha(x),\lim\limits_{x\to x_0}\alpha(x)=0$。 - -\textbf{例题:}如果$\lim\limits_{x\to 0}\dfrac{x-\sin x+f(x)}{x^4}$存在,则$\lim\limits_{x\to 0}\dfrac{x^3}{f(x)}$为常数多少? - -由$\lim\limits_{x\to 0}\dfrac{x\sin x+f(x)}{x^4}=A$脱帽:$\dfrac{x\sin x+f(x)}{x^4}=A+\alpha$。 - -得到:$f(x)=Ax^4+\alpha\cdot x^4-(x-\sin x)$。 - -反代入:$\lim\limits_{x\to 0}\dfrac{f(x)}{x^3}=\lim\limits_{x\to 0}\dfrac{Ax^4+\alpha\cdot x^4-x+\sin x}{x^3}=0+0-\dfrac{1}{6}=-\dfrac{1}{6}$。 - -$\therefore \lim\limits_{x\to 0}\dfrac{x^3}{f(x)}=-6$。 - -\subsection{求参数} - -因为求参数类型的题目中式子是未知的,所以求导后也是未知的,所以一般不要使用洛必达法则,而使用泰勒展开。 - -一般极限式子右侧等于一个常数,或是表明高阶或低阶。具体的关系参考无穷小比阶。 - -在求参数的时候要注意与0的关系。\medskip - -\textbf{例题:}设$\lim\limits_{x\to 0}\dfrac{\ln(1+x)-(ax+bx^2)}{x^2}=2$,求常数a,b。 - -根据泰勒展开式:$x\to 0,\ln(1+x)=x-\dfrac{x^2}{x}+o(x^2)$,$x-\ln(1+x)\sim\dfrac{1}{2}x^2\sim 1-\cos x$。 - -$\lim\limits_{x\to 0}\dfrac{\ln(1+x)-(ax+bx^2)}{x^2}=2$ - -$=\lim\limits_{x\to 0}\dfrac{(1-a)x-\left(\dfrac{1}{2}+b\right)x^2+o(x^2)}{x^2}=2\neq 0$ - -$1-a=0;-\left(\dfrac{1}{2}+b\right)=2$\medskip - -$\therefore a=1;b=-\dfrac{5}{2}$。 - -\subsection{极限存在性} - -一般会给出带有参数的例子,并给定一个点指明在该点极限存在,求参数。 - -若该点极限存在,则该点两侧的极限都相等。\medskip - -\textbf{例题:}设函数$f(x)=\left\{\begin{array}{lcl} - \dfrac{\sin x(b\cos x-1)}{e^x+a}, & & x>0 \\ - \dfrac{\sin x}{\ln(1+3x)}, & & x<0 -\end{array} -\right.$在$x=0$处极限存在,则$a$,$b$分别为。 - -解:首先根据极限在$x=0$存在,且极限的唯一性。分段函数在0两侧的极限值必然相等。 - -$\because\lim\limits_{x\to 0^-}\dfrac{\sin x}{\ln(1+3x)}=\lim\limits_{x\to 0^-}\dfrac{\sin x}{3x}=\dfrac{1}{3}=\lim\limits_{x\to 0^+}\dfrac{\sin x(b\cos x-1)}{e^x+a}$。 - -\medskip - -又$\lim\limits_{x\to 0^+}\dfrac{\sin x(b\cos x-1)}{e^x+a}$的分母的$e^x$当$x\to 0^+$时$e^x\to 1$,假如$a\neq-1$,则$e^x+a\neq 0$,则为一个常数。 - -从而提取常数因子:$\lim\limits_{x\to 0^+}\dfrac{\sin x(b\cos x-1)}{e^x+a}=\dfrac{1}{1+a}\lim\limits_{x\to 0^+}\sin x(b\cos x-1)$,这时候$\sin x$是趋向0的,而$b\cos x-1$无论其中的$b$为何值都是趋向一个常数或0,这时候他们的乘积必然为无穷小,从而无法等于$\dfrac{1}{3}$这个常数。 - -$\therefore a=-1$,从而让极限式子变为一个商的形式:\medskip - -$\lim\limits_{x\to 0^+}\dfrac{\sin x(b\cos x-1)}{e^x+a}=\lim\limits_{x\to 0^+}\dfrac{\sin x(b\cos x-1)}{e^x-1}=\lim\limits_{x\to 0^+}\dfrac{\sin x(b\cos x-1)}{x}$\medskip - -$=\lim\limits_{x\to 0^+}b\cos x-1=b-1=\dfrac{1}{3}$\medskip - -$\therefore a=-1,b=\dfrac{4}{3}$。 - -\subsection{极限唯一性} - -若极限存在则必然唯一。 - -\textbf{例题:}设$a$为常数,$\lim\limits_{x\to 0}\left(\dfrac{e^{\frac{1}{x}}-\pi}{e^{\frac{2}{x}}+1}+a\cdot\arctan\dfrac{1}{x}\right)$存在,求出极限值。 - -因为求$x\to 0$,所以需要分两种情况讨论: - -\medskip - -$\lim\limits_{x\to 0^+}\left(\dfrac{e^{\frac{1}{x}}-\pi}{e^{\frac{2}{x}}+1}+a\cdot\arctan\dfrac{1}{x}\right)$ - -$=\lim\limits_{x\to 0^+}\left(\dfrac{e^{\frac{1}{x}}-\pi}{e^{\frac{2}{x}}+1}\right)+\lim\limits_{x\to 0^+}\left(a\cdot\arctan\dfrac{1}{x}\right)$ - -$=\lim\limits_{x\to 0^+}\left(\dfrac{0\cdot\left(e^{\frac{2}{x}}\right)^2+e^{\frac{1}{x}}-\pi}{1\cdot\left(e^{\frac{2}{x}}\right)^2+1}\right)+a\cdot\dfrac{\pi}{2}=a\cdot\dfrac{\pi}{2}$ - -\medskip - -$\lim\limits_{x\to 0^-}\left(\dfrac{e^{\frac{1}{x}}-\pi}{e^{\frac{2}{x}}+1}+a\cdot\arctan\dfrac{1}{x}\right)=-\pi+a\cdot\left(-\dfrac{\pi}{2}\right)=-\pi-\dfrac{\pi}{2}\cdot a$ - -因为极限值具有唯一性,所以$-\pi-\dfrac{\pi}{2}a=\dfrac{\pi}{2}a$,所以$a=-1$,极限值为$-\dfrac{\pi}{2}$。 - -\subsection{函数连续性} - -函数的连续性代表:极限值=函数值。 - -\subsubsection{判断函数连续} - -题目给出函数,往往是分段函数,然后判断分段点的连续性。\medskip - -\textbf{例题:}讨论函数$f(x)=\left\{\begin{array}{lcl} - \left[\dfrac{(1+x)^{\frac{1}{x}}}{e}\right]^{\frac{1}{x}},& & x>0 \\ - e^{-\frac{1}{2}}, & & x\leqslant 0 -\end{array}\right.$在$x=0$处的连续性。 - -因为$\lim\limits_{x\to 0^+}f(x)=\lim\limits_{x\to 0^+}\left[\dfrac{(1+x)^{\frac{1}{x}}}{e}\right]^{\frac{1}{x}}=e^{\lim\limits_{x\to 0^+}\frac{1}{x}\ln[\frac{(1+x)^{\frac{1}{x}}}{e}]}$。 - -又$\lim\limits_{x\to 0^+}\dfrac{1}{x}\ln\left[\dfrac{(1+x)^{\frac{1}{x}}}{e}\right]=\lim\limits_{x\to 0^+}\dfrac{1}{x}\left[\dfrac{1}{x}\ln(1+x)-1\right]=\lim\limits_{x\to 0^+}\dfrac{\ln(1+x)-x}{x^2}$ - -$=\lim\limits_{x\to 0^+}\dfrac{\dfrac{1}{1+x}-1}{2x}=\lim\limits_{x\to 0^+}-\dfrac{1}{2(1+x)}=-\dfrac{1}{2}$。 - -$\therefore\lim\limits_{x\to 0^+}f(x)=e^{-\frac{1}{2}}$。 - -又$\lim\limits_{x\to 0^-}f(x)=\lim\limits_{x\to 0^-}e^{-\frac{1}{2}}=e^{-\frac{1}{2}}$,且$f(0)=e^{-\frac{1}{2}}$。 - -从而$\lim\limits_{x\to 0^+}f(x)=\lim\limits_{x\to 0^-}f(x)=f(0)$,所以$f(x)$在$x=0$处连续。 - -\subsubsection{连续性求极限} - -\textbf{例题:}函数在$f(x)$在$x=1$处连续,且$f(1)=1$,求$\lim\limits_{x\to+\infty}\ln\left[2+f\left(x^{\frac{1}{x}}\right)\right]$。 - -根据题目,所求的$\lim\limits_{x\to+\infty}\ln\left[2+f\left(x^{\frac{1}{x}}\right)\right]$中,唯一未知的且会随着$x\to+\infty$而变换就是$f\left(x^{\frac{1}{x}}\right)$。如果我们可以求出这个值就可以了。 - -而我们对于$f(x)$的具体的关系是未知的,只知道$f(1)=1$。那么先需要考察$\lim\limits_{x\to+\infty}x^{\frac{1}{x}}$的整数最大值。 - -$\lim\limits_{x\to+\infty}x^{\frac{1}{x}}=e^{\lim\limits_{x\to+\infty}\frac{\ln x}{x}}=e^{\lim\limits_{x\to+\infty}\frac{1}{x}}=e^0=1$ - -$\therefore\lim\limits_{x\to+\infty}f(x^{\frac{1}{x}})=f(1)=1$。 - -\subsection{迭代式数列} - -\subsubsection{数列表达式} - -最重要的是将迭代式进行变形。 - -\textbf{例题:}数列$\{a_n\}$满足$a_0=0,a_1=1,2a_{n+1}=a_n+a_{n-1},n=1,2,\cdots$。计算$\lim\limits_{n\to\infty}a_n$。 - -首先看题目,给出的递推式设计到二阶递推,即存在三个数列变量,所以我们必须先求出对应的数列表达式。因为这个表达式涉及三个变量,所以尝试对其进行变型: - -$a_{n+1}-a_n=\dfrac{a_{n-1}-a_n}{2}=\left(-\dfrac{1}{2}\right)(a_n-a_{n-1})=\left(-\dfrac{1}{2}\right)^2(a_{n-1}-a_{n-2})$ - -$=\cdots$ - -$=\left(-\dfrac{1}{2}\right)^n(a_1-a_0)=\left(-\dfrac{1}{2}\right)^n$ - -然后得到了$a_{n+1}-a_n=\left(-\dfrac{1}{2}\right)^n$,而需要求极限,所以使用列项相消法的逆运算: - -$a_n=(a_n-a_{n-1})+(a_{n-1}-a_{n-2})+\cdots+(a_1-a_0)+a_0$\medskip - -$=\left(-\dfrac{1}{2}\right)^{n-1} + \left(-\dfrac{1}{2}\right)^{n-2} + \cdots + \left(-\dfrac{1}{2}\right)^0$\medskip - -$=\dfrac{1\cdot\left(1-\left(-\dfrac{1}{2}\right)^n\right)}{1-\left(-\dfrac{1}{2}\right)}=\dfrac{2}{3}\left[1-\left(-\dfrac{1}{2}\right)^n\right]$\medskip - -$\therefore\lim\limits_{n\to\infty}a_n=\dfrac{2}{3}$ - -\subsubsection{单调有界准则} - -对于无法将关系式通过变形归纳为一般式的关系式,对于其极限就必须使用单调有界准则来求出。 - -单调有界的数列必有极限。需要证明单调性和有界性,然后对式子求极限就能求出目标极限。 - -\textbf{例题:}$x_0=0$,$x_n=\dfrac{1+2x_{n-1}}{1+x_{n-1}}(n\in N*)$,求$\lim\limits_{n\to\infty}x_n$。\medskip - -首先应该知道数列的趋向都是趋向正无穷。 - -然后对关系式进行变形:$x_n=\dfrac{1+2x_{n-1}}{1+x_{n-1}}=1+\dfrac{x_{n-1}}{1+x_{n-1}}=2-\dfrac{1}{1+x_{n-1}}$。 - -首先证明单调性,令$f(x)=2-\dfrac{1}{1+x}$。 - -$\therefore f'(x)=\dfrac{1}{(x+1)^2}>0$,则$f(x)$单调递增。 - -所以不管$x=x_{n-1}$或其他,$f'(x)>0$,$x_n$都是单调递增,则$x_n\geqslant x_0=0$。 - -然后证明有界性,$\because x_n\geqslant 0$且单调,$\therefore x_n=2-\dfrac{1}{1+x_{n-1}}\in[0,2]$。 - -从而$x_n$有界。 - -所以根据单调有界定理,$x_n$的极限存在。 - -对于关系式两边取极限: - -$\lim\limits_{n\to\infty}x_n=\lim\limits_{n\to\infty}\dfrac{1+2x_{n-1}}{1+x_{n-1}}=\dfrac{1+2\lim\limits_{n\to\infty}x_{n-1}}{1+\lim\limits_{n\to\infty}x_{n-1}}=\dfrac{1+2\lim\limits_{n\to\infty}x_n}{1+\lim\limits_{n\to\infty}x_n}$。 - -解该一元二次方程:$\lim\limits_{n\to\infty}x_n=\dfrac{1\pm\sqrt{5}}{2}$,又根据保号性,$\lim\limits_{n\to\infty}x_n>0$。 - -$\therefore\lim\limits_{n\to\infty}x_n=\dfrac{1+\sqrt{5}}{2}$。 - -而许多题目只给出样子,连通项公式都不会给出。\medskip - -\textbf{例题:}求出数列$\sqrt{2}$,$\sqrt{2+\sqrt{2}}$,$\sqrt{2+\sqrt{2+\sqrt{2}}}$$\cdots$的极限。 - -根据数列样式,无法通过普通的通项公式来表达,所以需要考虑使用递推式来表示:$x_{n+1}=\sqrt{2+x_n}$。 - -首先证明有界性: - -给定一个任意的正整数$k$,再根据递推式,假定$x_k<2$,所以$x_{k+1}=\sqrt{2+x_k}<\sqrt{2+2}=2$。且$x_1=\sqrt{2}$满足假定,所以$x_k<2$对于任意的正整数$k$都成立,所以$x_n$存在上界2。 - -然后证明单调性,根据其递推式: - -$x_{n+1}-x_n=\sqrt{2+x_n}-x_n=\dfrac{2+x_n-x_n^2}{\sqrt{2+x_n}+x_n}=\dfrac{-(x_n-2)(x_n+1)}{\sqrt{2+x_n}+x_n}$。\medskip - -又$0b>0$,证明:$\dfrac{a-b}{a}<\ln\dfrac{a}{b}<\dfrac{a-b}{b}$。 + +因为$\ln\dfrac{a}{b}=\ln a-\ln b$,所以令$f(x)=\ln x$。 + +所以根据拉格朗日中值定理:$\ln a-\ln b=f'(\xi)(a-b)$($\xi\in(b,a)$)。 + +又$f'(\xi)=\dfrac{1}{\xi}$,所以$\ln a-\ln b=\dfrac{a-b}{\xi}$。 + +又$\xi\in(b,a)$,所以$\dfrac{1}{\xi}\in(\dfrac{1}{a},\dfrac{1}{b})$。 + +所以$\dfrac{a-b}{a}<\dfrac{a-b}{\xi}<\dfrac{a-b}{b}$,从而$\dfrac{a-b}{a}<\ln\dfrac{a}{b}<\dfrac{a-b}{b}$,得证。 + +\paragraph{查找特定值} \leavevmode \medskip + +对于证明一种不等式,如果里面没有差式,也无法转换为差式,那么就可以考虑制造差式,对于$f(x)$一般选择更高阶的,$a$选择$x$,$b$要根据题目和不等式设置一个常数。 + +一般是0或1。可以先尝试1。 + +\textbf{例题:}当$x>1$时,证明$e^x>ex$。 + +题目中没有差式,所以需要选择一个函数作为基准函数,里面有一个指数函数和一个幂函数,所以选择$e^x$作为基准函数。 + +然后选择一个常数作为$b$值,可以先选一个1作为$b$值:$f(x)-f(1)=f'(\xi)(x-1)$。 + +从而$e^x-e=e^\xi(x-1)$,$\xi\in(1,x)$,所以$e^x-e>e(x-1)$,即$e^x>ex$,得证。 + +\subsubsection{柯西中值定理} + +需要找到两个函数,使得$\dfrac{f(b)-f(a)}{F(b)-F(a)}=\dfrac{f'(\xi)}{F'(\xi)}$。 + +\textbf{例题:}设$00$时,$x\to 0^+$,两边都乘以10,$10-x0$时,$x\to 0^-$,同样也是夹逼准则得到极限为10。\medskip + +$\therefore \lim\limits_{x\to 0}x\left[\dfrac{10}{x}\right]=10$。 + +\subsubsection{拉格朗日中值定理} + +对于形如$f(a)-f(b)$的极限式子就可以使用拉格朗日中值定理,这个$f(x)$为任意的函数。使用拉格朗日中值定理最重要的还是找到这个$f(x)$。 + +可以将极限式子中形如$f(a)-f(b)$的极限部分使用拉格朗日中值定理进行替换。 + +\textbf{例题:}求极限$\lim\limits_{n\to\infty}n^2\left(\arctan\dfrac{2}{n}-\arctan\dfrac{2}{n+1}\right)$。\medskip + +因为式子不算非常复杂,其实也可以通过洛必达法则来完成,但是求导会很复杂。而$\arctan x$可以认定为$f(x)$。 + +从而$\arctan\dfrac{2}{n}-\arctan\dfrac{2}{n+1}$为$f(\dfrac{2}{n})-f(\dfrac{2}{n+1})=f'(\xi)\left(\dfrac{2}{n}-\dfrac{2}{n+1}\right)$。 + +其中$\dfrac{2}{n+1}<\xi<\dfrac{2}{n}$,而当$n\to\infty$时,$f'(\xi)=\dfrac{1}{1+\xi^2}\to 1$。 + +$\therefore\arctan\dfrac{2}{n}-\arctan\dfrac{2}{n+1}\sim\dfrac{2}{n}-\dfrac{2}{n+1}=\dfrac{2}{n(n+1)}$。 + +$\therefore\lim\limits_{n\to\infty}n^2\left(\arctan\dfrac{2}{n}-\arctan\dfrac{2}{n+1}\right)=\lim\limits_{n\to\infty}n^2\cdot\dfrac{2}{n(n+1)}=2$。 + +\subsubsection{洛必达法则} + +洛必达法则的本质是降低商形式的极限式子的幂次。 + +洛必达在处理一般的极限式子比较好用,但是一旦式子比较复杂最好不要使用洛必达法则,最好是对求导后有规律或幂次较低的式子进行上下求导。 + +对于幂次高的式子必然使用洛必达法则。 + +洛必达法则必须使用在分式都趋向0或$\infty$时,如果不是这样的趋向则不能使用。如: + +\textbf{例题:}求$\lim\limits_{x\to 1}\dfrac{x^2-x+1}{(x-1)^2}$。 + +如果使用洛必达法则,则会得到结果为1,这是错误的,因为分子在$x\to 1$时结果为常数1。正确的计算方式: + +$=\lim\limits_{x\to 1}\dfrac{1}{(x-1)^2}=\infty$。 + +\subsubsection{泰勒公式} + +泰勒公式一般会使用趋向0的麦克劳林公式,且一般只作为极限计算的一个小部分,用来替代一个部分。 + +且一般只有麦克劳林公式表上的基本初等函数才会使用倒泰勒公式,复合函数最好不要使用。 + +\textbf{例题:}求极限$\lim\limits_{x\to 0}\dfrac{\arcsin x-\arctan x}{\sin x-\tan x}$。\medskip + +分析:该题目使用洛必达法则会比较麻烦且难以计算,所以先考虑是否能用泰勒展开: + +$x\to 0$,$\sin x=x-\dfrac{1}{6}x^3+o(x^3)$,$\tan x=x+\dfrac{1}{3}x^3+o(x^3)$,$\arcsin x=x+\dfrac{1}{6}x^3+o(x^3)$,$\arctan x=x-\dfrac{1}{3}x^3+o(x^3)$。 + +$\therefore \sin x-\tan x=-\dfrac{1}{2}x^3+o(x^3)$,$\arcsin x-\arctan x=\dfrac{1}{2}x^3+o(x^3)$ + +$\therefore \text{原式}=\dfrac{\dfrac{1}{x}x^3+o(x^3)}{-\dfrac{1}{2}x^3+o(x^3)}=-1$。 + + +\subsection{常用化简技巧} + +\subsubsection{对数法则} + +如$\log_n(a\cdot b)=\log_n a+\log_n b$,$\log_n\dfrac{a}{b}=\log_na-\log_nb$。 + +换底公式:对于$a,c\in(0,1)\cup(1,+\infty)$且$b\in(0,+\infty)$时,有$\log_ab=\dfrac{\log_nb}{\log_na}$。 + +\textbf{例题:}求$\lim\limits_{x\to 0}\dfrac{(e^{x^2}-1)(\sqrt{1+x}-\sqrt{1-x})}{[\ln(1-x)+\ln(1+x)]\sin\dfrac{x}{x+1}}$。 + +注意在积或商的时候不能把对应的部分替换为0,如分母部分的$[\ln(1-x)+\ln(1+x)]$就无法使用$\ln(1+x)\sim x$替换为$-x+x$,这样底就是0了,无法求得最后的极限。 + +这时可以尝试变形,如对数函数相加等于对数函数内部式子相乘:$\ln(1-x)+\ln(1+x)=\ln(1-x^2)\sim-x^2$。 + +\subsubsection{指数法则} + +当出现$f(x)^{g(x)}$的类似幂函数与指数函数类型的式子,需要使用$u^v=e^{v\ln u}$。 + +一般需要与洛必达法则配合使用。 + +\textbf{例题:}求$\lim\limits_{x\to+\infty}(x+\sqrt{1+x^2})^{\frac{1}{x}}$。 + +$\lim\limits_{x\to+\infty}(x+\sqrt{1+x^2})^{\frac{1}{x}}=e^{\lim\limits_{x\to+\infty}\frac{\ln(x+\sqrt{1+x^2})}{x}} \left(\ln(x+\sqrt{1+x^2})'=\dfrac{1}{\sqrt{1+x^2}}\right)$\medskip + +$=e^{\lim\limits_{x\to+\infty}\frac{1}{\sqrt{1+x^2}}}=e^0=1$ + +\textbf{例题:}求$\lim\limits_{x\to 0}\left(\dfrac{a^x+b^x+c^x}{3}\right)^{\frac{1}{x}}(a>0,b>0,c>0)$。\medskip + +$=e^{\lim\limits_{x\to 0}\frac{\ln\left(\frac{a^x+b^x+c^x}{3}\right)}{x}}=e^{\lim\limits_{x\to 0}\frac{\ln(a^x+b^x+c^x)-\ln 3}{x}}=e^{\lim\limits_{x\to 0}\frac{a^x\ln a+b^x\ln b+c^x\ln c}{a^x+b^x+c^x}}$(洛必达法则) + +$=e^{\lim\limits_{x\to 0}\frac{\ln a+\ln b+\ln c}{1+1+1}}=e^{\lim\limits_{x\to 0}\frac{\ln(abc)}{3}}=\sqrt[3]{abc}$。\medskip + +\textbf{例题:}求$\lim\limits_{n\to\infty}n\left[\left(1+\dfrac{1}{n}\right)^{\frac{n}{2}}-\sqrt{e}\right]$。 + +首先对于幂指函数需要取指数,所以$\left(1+\dfrac{1}{n}\right)^{\frac{n}{2}}=e^{\frac{n}{2}\ln(1+\frac{1}{n})}$。\medskip + +而后面的多一个$\sqrt{e}$导致整个式子变为一个复杂的式子,而与$e^x$相关的是$e^x-1\sim x$。 + +所以$e^{\frac{n}{2}\ln(1+\frac{1}{n})}-\sqrt{e}=e^{\frac{1}{2}}\cdot\left(e^{\frac{n}{2}\ln(1+\frac{1}{n})-\frac{1}{2}}-1\right)=e^{\frac{1}{2}}\cdot\left[\dfrac{n}{2}\ln(1+\dfrac{1}{n})-\dfrac{1}{2}\right]$。 + +综上: + +$\lim\limits_{n\to\infty}n\left[\left(1+\dfrac{1}{n}\right)^{\frac{n}{2}}-\sqrt{e}\right]=\lim\limits_{n\to\infty}n\left(e^{\frac{n}{2}\ln(1+\frac{1}{n})}-\sqrt{e}\right)$ \medskip + +$=\lim\limits_{n\to\infty}n\left[e^{\frac{1}{2}}\cdot\left(e^{\frac{n}{2}\ln(1+\frac{1}{n})-\frac{1}{2}}-1\right)\right]=\dfrac{e^{\frac{1}{2}}}{2}\lim\limits_{n\to\infty}n^2\left[\ln\left(1+\frac{1}{n}\right)-\dfrac{1}{n}\right]$ + +$=\dfrac{e^{\frac{1}{2}}}{2}\lim\limits_{n\to\infty}\dfrac{\dfrac{1}{n}-\dfrac{1}{2n^2}-\dfrac{1}{n}}{\dfrac{1}{n^2}}=\dfrac{e^{\frac{1}{2}}}{2}\cdot\left(-\dfrac{1}{2}\right)=-\dfrac{\sqrt{e}}{4}$ + +\subsubsection{三角函数关系式} + +\textbf{例题:}求极限$\lim\limits_{x\to 0}\left(\dfrac{1}{\sin^2x}-\dfrac{\cos^2x}{x^2}\right)$。\medskip + +$\lim\limits_{x\to 0}\left(\dfrac{1}{\sin^2x}-\dfrac{\cos^2x}{x^2}\right)=\lim\limits_{x\to 0}\dfrac{x^2-\sin^2x\cos^2x}{\sin^2x\cdot x^2} (\sin x\sim x)$ \medskip + +$=\lim\limits_{x\to 0}\dfrac{x^2-\sin^2x\cos^2x}{x^4} (\sin x\cos x\sim\dfrac{1}{2}\sin 2x)=\lim\limits_{x\to 0}\dfrac{x^2-\dfrac{1}{4}\sin^22x}{x^4}$ \medskip + +$=\lim\limits_{x\to 0}\dfrac{2x-\dfrac{1}{4}\cdot 2\sin 2x\cdot\cos 2x\cdot 2}{4x^3} (\sin x\cos x\sim\dfrac{1}{2}\sin 2x)=\lim\limits_{x\to 0}\dfrac{2x-\dfrac{1}{2}\sin 4x}{4x^3}$ \medskip + +$=\lim\limits_{x\to 0}\dfrac{2-\dfrac{1}{2}\cos 4x\cdot 4}{12x^2}=\dfrac{1}{6}\lim\limits_{x\to 0}\dfrac{1-\cos 4x}{x^2} (1-\cos x\sim \dfrac{1}{2}x^2)=\dfrac{4}{3}$ + +\subsubsection{提取常数因子} + +提取常数因子就是提取出能转换为常数的整个极限式子的因子。这个因子必然在自变量的趋向时会变为非0的常数,那么这个式子就可以作为常数提出。 + +\subsubsection{提取公因子} + +当作为商的极限式子上下都具有公因子时可以提取公因子然后相除,从而让未知数集中在分子或分母上。 + +\textbf{例题:}求$\lim\limits_{x\to 4}\dfrac{x^2-6x+8}{x^5-5x+4}$。 + +需要先提取公因子: + +$=\lim\limits_{x\to 4}\dfrac{(x-2)(x-4)}{(x-1)(x-4)}=\lim\limits_{x\to 4}\dfrac{x-2}{x-1}=\dfrac{2}{3}$。 + +(当然可以使用洛必达法则得到极限为$\lim\limits_{x\to 4}\dfrac{2x-6}{2x-5}=\lim\limits_{x\to 4}\dfrac{8-6}{8-5}$) + +\textcolor{orange}{注意:}提取公因子的时候应该注意开平方等情况下符号的问题。如果极限涉及倒正负两边则必须都讨论。 + +当趋向为负且式子中含有根号的时候最好提取负因子,从而让趋向变为正。\medskip + +\textbf{例题:}求$\lim\limits_{x\to-\infty}\left[\sqrt{4x^2+x}\ln\left(2+\dfrac{1}{x}\right)+2\ln 2x\right]$。\medskip + +题目的形式为$\infty-\infty$,所以必须使用后面的倒代换转换为商的形式。\medskip + +$=\lim\limits_{x\to-\infty}-x\left[\sqrt{4+\dfrac{1}{x}}\ln\left(2+\dfrac{1}{x}\right)-2\ln 2\right]$。 \medskip + +这里就需要注意到因为$\sqrt{4x^2+x}$的限制导致这个式子必然为正数,而$x\to-\infty$代表自变量为负数,所以提出来的$x$必然是负数,而原式是正数,所以就需要添加一个负号,而后面的$2\ln 2x$则没有要求,所以直接变成$-2\ln 2$就可以了。 + +将$x$下翻变成分母为$\dfrac{1}{x}$,并令$t=\dfrac{1}{x}$。\medskip + +$=\lim\limits_{t\to 0^-}\dfrac{\sqrt{t+4}\ln\left(2+\dfrac{1}{x}\right)-2\ln 2}{-t}$。\medskip + +幂次不高可以尝试洛必达:\medskip + +$=\lim\limits_{t\to 0^-}\left(\dfrac{1}{2}\cdot\dfrac{\ln(2+t)}{\sqrt{t+4}}+\dfrac{\sqrt{t+4}}{2+t}\right)=-\left(\dfrac{1}{2}\cdot\dfrac{\ln 2}{2}+\dfrac{2}{2}\right)=-\dfrac{\ln 2}{4}-1$。 + +\subsubsection{有理化} + +当遇到带有根号的式子可以使用等价无穷小,但是只针对形似$(1+x)a-1\sim ax$的式子,而针对$x^a\pm x^b$的式子则无法替换,必须使用有理化来将单个式子变为商的形式。 + +如$\sqrt{a}\pm\sqrt{b}=\dfrac{a+b}{\sqrt{a}\mp\sqrt{b}}$。\medskip + +\textbf{例题:}求极限$\lim\limits_{x\to-\infty}x(\sqrt{x^2+100}+x)$。 + +首先定性分析:$\lim\limits_{x\to-\infty}x\cdot(\sqrt{x^2+100}+x)$。 + +在$x\to-\infty$趋向时,$x$就趋向无穷大。 + +而$\sqrt{x^2+100}$为一次,所以$\sqrt{x^2+100}+x$趋向0。 + +又$\sqrt{x^2+100}$在$x\to-\infty$时本质为根号差,所以有理化: + +$\lim\limits_{x\to-\infty}x(\sqrt{x^2+100}+x)=\lim\limits_{x\to-\infty}x\dfrac{x^2+100-x^2}{\sqrt{x^2+100}-x}=\lim\limits_{x\to-\infty}\dfrac{100x}{\sqrt{x^2+100}-x}$\medskip + +$\xRightarrow{\text{令}x=-t}\lim\limits_{t\to+\infty}\dfrac{-100t}{\sqrt{t^2+100}+t}=\lim\limits_{t\to+\infty}\dfrac{-100}{\sqrt{1+\dfrac{100}{t^2}}+1}=-50$\medskip + +\textbf{例题:}求$\lim\limits_{x\to 0}\dfrac{\sqrt{1+\tan x}-\sqrt{1+\sin x}}{x\sqrt{1+\sin^2x}-x}$。 + +$=\lim\limits_{x\to 0}\sqrt{1+\tan x}-\sqrt{1+\sin x}\cdot\lim\limits_{x\to 0}\dfrac{1}{x\sqrt{1+\sin^2x}-x}$\medskip + +$=\lim\limits_{x\to 0}\dfrac{\tan x-\sin x}{\sqrt{1+\tan x}+\sqrt{1+\sin x}}\cdot\lim\limits_{x\to 0}\dfrac{x\sqrt{1+\sin^2x}+x}{x^2(1+\sin^2x)-x^2}$\medskip + +$=\lim\limits_{x\to 0}\dfrac{\tan x-\sin x}{\sqrt{1+\tan x}+\sqrt{1+\sin x}}\cdot\lim\limits_{x\to 0}\dfrac{\sqrt{1+\sin^2x}+1}{x\sin^2x}$\medskip + +$=\lim\limits_{x\to 0}\dfrac{\tan x-\sin x}{2}\cdot\lim\limits_{x\to 0}\dfrac{2}{x\sin^2x}$ + +$=\lim\limits_{x\to 0}\dfrac{\tan x-\sin x}{x\sin^2x}=\lim\limits_{x\to 0}\dfrac{1-\cos x}{x\cos x\sin x}=\lim\limits_{x\to 0}\dfrac{\dfrac{1}{2}x^2}{x^2}=\dfrac{1}{2}$。 + +\subsubsection{换元法} + +换元法本身没什么技巧性,主要是更方便计算。最重要的是获取到共有的最大因子进行替换。 + +\textbf{例题:}求极限$\lim\limits_{x\to 1^-}\ln x\ln(1-x)$。 + +当$x\to 1^-$时,$\ln x$趋向0,$\ln(1-x)$趋向$-\infty$。 + +又$x\to 0$,$\ln(1+x)\sim x$,所以$x\to 1$,$\ln x\sim x-1$: + +$\lim\limits_{x\to 1^-}\ln x\ln(1-x)=\lim\limits_{x\to 1^-}(x-1)\ln(1-x)\xRightarrow{令t=1-x} =-\lim\limits_{t\to 0^+}t\ln t$ + +$=-\lim\limits_{t\to 0^+}\dfrac{\ln t}{\dfrac{1}{t}}=-\lim\limits_{t\to 0^+}\dfrac{\dfrac{1}{t}}{-\dfrac{1}{t^2}}=\lim\limits_{t\to 0^+}t=0$ + +\subsubsection{倒代换} + +\paragraph{含有分式} \leavevmode \medskip + +当极限式子中含有分式中一般都需要用其倒数,把分式换成整式方便计算。 + +\textbf{例题:}求极限$\lim\limits_{x\to 0}\dfrac{e^{-\frac{1}{x^2}}}{x^{100}}$ \medskip + +$\lim\limits_{x\to 0}\dfrac{e^{-\frac{1}{x^2}}}{x^{100}}=\lim\limits_{x\to 0}\dfrac{e^{-\frac{1}{x^2}}\cdot 2x^{-3}}{100x^{99}}=\lim\limits_{x\to 0}\dfrac{1}{50}\lim\limits_{x\to 0}\dfrac{e^{-\frac{1}{x^2}}}{x^{102}}$ + +\medskip + +使用洛必达法则下更复杂,因为分子的幂次为负数,导致求导后幂次绝对值越来越大,不容易计算。 + +使用倒代换再洛必达降低幂次,令$t=\dfrac{1}{x^2}$ + +$\lim\limits_{x\to 0}\dfrac{e^{-\frac{1}{x^2}}}{x^{100}}=\lim\limits_{t\to+\infty}\dfrac{e^{-t}}{t^{-50}}=\lim\limits_{t\to+\infty}\dfrac{t^{50}}{e^t}=\lim\limits_{t\to+\infty}\dfrac{50t^{49}}{e^t}$ + +$=\cdots$ + +$=\lim\limits_{t\to+\infty}\dfrac{50!}{e^t}=0$ + +\textbf{例题:}求极限$\lim\limits_{x\to+\infty}[x^2(e^{\frac{1}{x}}-1)-x]$。 + +该式子含有分数,所以尝试使用倒数代换:\medskip + +$\lim\limits_{x\to+\infty}[x^2(e^{\frac{1}{x}}-1)-x]\xRightarrow{\text{令}x=\frac{1}{t}}\lim\limits_{t\to 0^+}\left(\dfrac{e^t-1}{t^2}-\dfrac{1}{t}\right)=\lim\limits_{t\to 0^+}\dfrac{e^t-1-t}{t^2}$ + +$\xRightarrow{\text{泰勒展开}e^t}\lim\limits_{t\to 0^+}\dfrac{\dfrac{1}{2}t^2}{t^2}=\dfrac{1}{2}$ + +\paragraph{\texorpdfstring{$\infty-\infty$}\ 型} \leavevmode \medskip + +\paragraph{\texorpdfstring{$\infty\cdot\infty$}\ 型} \leavevmode \medskip + +\subsubsection{拆项} + +拆项需要根据式子形式进行,所以很难找到普遍规律。 + +\paragraph{积拆项} \leavevmode \medskip + +\textbf{例题:}求$\lim\limits_{n\to\infty}\dfrac{(n+1)(n+2)(n+3)\cdots(n+6)}{6n^6}$。 + +需要将分子和分母都拆为6项: + +$=\dfrac{1}{6}\lim\limits_{n\to\infty}\dfrac{n+1}{n}\times\dfrac{n+2}{n}\times\cdots\dfrac{n+6}{n}=\dfrac{1}{6}\lim\limits_{n\to\infty}(1+\dfrac{1}{n})(1+\dfrac{2}{n})\cdots(1+\dfrac{6}{n})=\dfrac{1}{6}$。 + +当极限式子中出现不知道项数的$n$时,一般需要使用拆项,把项重新组合。一般的组合是根据等价无穷小。 + +\paragraph{和拆项} \leavevmode \medskip + +而对于复杂的具有同一结构的和的式子也可以考虑拆项。 + +\textbf{例题:}求极限$\lim\limits_{x\to 0}\left(\dfrac{e^x+e^{2x}+\cdots+e^{nx}}{n}\right)^{\frac{e}{x}}$。($n\in N^+$) + +这里可以使用等价无穷小$e^x-1\sim x$。 + +$\lim\limits_{x\to 0}\left(\dfrac{e^x+e^{2x}+\cdots+e^{nx}}{n}\right)^{\frac{e}{x}}=e^{\lim\limits_{x\to 0}\frac{e}{x}\ln\left(\frac{e^x+e^{2x}+\cdots+e^{nx}}{n}\right)}$ + +$=e^{\lim\limits_{x\to 0}\frac{e}{x}\left(\frac{e^x+e^{2x}+\cdots+e^{nx}}{n}-1\right)}=e^{\lim\limits_{x\to 0}\frac{e}{x}\left(\frac{e^x+e^{2x}+\cdots+e^{nx}-n}{n}\right)}$ + +$=e^{\frac{e}{n}\lim\limits_{x\to 0}\left(\frac{e^x-1}{x}+\frac{e^{2x}-1}{x}+\cdots+\frac{e^{nx}-1}{x}\right)}=e^{\frac{e}{n}[1+2+\cdots+n]}=e^{\frac{e}{n}\cdot\frac{n(1+n)}{2}}=e^{\frac{e(1+n)}{2}}$ + +\textbf{例题:}求$\lim\limits_{x\to 0}\dfrac{1-\cos x\sqrt{\cos 2x}\sqrt[3]{\cos 3x}}{\ln\cos x}$\medskip + +可以使用$\cos x-1\sim\dfrac{x^2}{2}$。\medskip + +$\lim\limits_{x\to 0}\dfrac{1-\cos x\sqrt{\cos 2x}\sqrt[3]{\cos 3x}}{\ln\cos x}$ + +$=\lim\limits_{x\to 0}\dfrac{1-\cos x+\cos x-\cos x\sqrt{\cos 2x}\sqrt[3]{\cos 3x}}{-\dfrac{x^2}{2}}$ + +$=\lim\limits_{x\to 0}\dfrac{\dfrac{x^2}{2}}{-\dfrac{x^2}{2}}+\dfrac{\cos x(1-\sqrt{\cos 2x}\sqrt[3]{\cos 3x})}{-\dfrac{x^2}{2}}$ + +$=-1+\lim\limits_{x\to 0}\dfrac{(1-\sqrt{\cos 2x})+\sqrt{\cos 2x}-\sqrt{\cos 2x}\sqrt[3]{\cos 3x}}{-\dfrac{x^2}{2}}$ + +$=-1+\lim\limits_{x\to 0}\dfrac{-\dfrac{1}{2}(\cos 2x-1)+\sqrt{\cos 2x}(1-\sqrt[3]{\cos 3x})}{-\dfrac{x^2}{2}}$ + +$=-1+\lim\limits_{x\to 0}\dfrac{-\dfrac{1}{2}(-\dfrac{4x^2}{2})+\left(-\dfrac{1}{3}\right)\left(-\dfrac{9x^2}{2}\right)}{-\dfrac{x^2}{2}}=-6$ + +\subsection{极限计算形式} + +极限相关计算形式主要分为下面六种: + +\begin{enumerate} + \item 未定式:直接根据式子计算极限值。 + \item 极限转换:根据已知的极限值计算目标极限值。 + \item 求参数:已知式子的极限值,计算式子中未知的参数。 + \item 极限存在性:根据式子以及极限存在性计算极限或参数。 + \item 极限唯一性:式子包含参数,根据唯一性计算两侧极限并求出参数与极限值。 + \item 函数连续性:根据连续性与附加条件计算极限值或参数。 + \item 迭代式数列:根据数列迭代式计算极限值。 + \item 变限积分:根据变限积分计算极限值。 +\end{enumerate} + +\subsubsection{极限不定式类型} + +七种:$\dfrac{0}{0},\dfrac{\infty}{\infty},0\cdot\infty,\infty-\infty,\infty^0,0^0,1^\infty$。 + +\medskip + +\ding{172}其中$\dfrac{0}{0}$为洛必达法则的基本型。$\dfrac{\infty}{\infty}$可以类比$\dfrac{0}{0}$的处理方式。$0\cdot\infty$可以转为$\dfrac{0}{\dfrac{1}{\infty}}=\dfrac{0}{0}=\dfrac{\infty}{\dfrac{1}{0}}=\dfrac{\infty}{\infty}$。设置分母有原则,简单因式才下放(简单:幂函数,e为底的指数函数)。 + +\medskip +6 +\ding{173}$\infty-\infty$可以提取公因式或通分,即和差化积。 + +\medskip + +\ding{174}$\infty^0,0^0,1^\infty$,就是幂指函数。 + +\medskip + +$ +u^v=e^{v\ln u}=\left\{ +\begin{array}{lcl} + \infty^0 & \rightarrow & e^{0\cdot+\infty} \\ + 0^0 & \rightarrow & e^{0\cdot-\infty} \\ + 1^\infty & \rightarrow & e^{\infty\cdot 0} \\ +\end{array} \right. +$ + +\medskip + +$\therefore \lim u^v=e^{\lim v\cdot\ln u}=e^{\lim v(u-1)}$ + +综上,无论什么样的四则形式,都必须最后转换为商的形式。 + +\subsubsection{极限转换} + +\paragraph{整体换元} \leavevmode \medskip + +最常用的方式就是将目标值作为一个部分,然后对已知的式子进行替换。 + +\textbf{例题:}已知$\lim\limits_{x\to 0}\dfrac{\ln(1-x)+xf(x)}{x^2}=0$,求$\lim\limits_{x\to 0}\dfrac{f(x)-1}{x}$。\medskip + +令目标$\dfrac{f(x)-1}{x}=t$,$\therefore f(x)=tx+1$。\medskip + +$\lim\limits_{x\to 0}\dfrac{\ln(1-x)+xf(x)}{x^2}=\lim\limits_{x\to 0}\dfrac{\ln(1-x)+tx^2+x}{x^2} (\text{泰勒展开})$\medskip + +$=\lim\limits_{x\to 0}\dfrac{-x-\dfrac{x^2}{2}+tx^2+x}{x^2}=\lim\limits_{x\to 0}\dfrac{\left(t-\dfrac{1}{2}\right)x^2}{x^2}=\lim\limits_{x\to 0}\left(t-\dfrac{1}{2}\right)=0$ + +$\therefore\lim\limits_{x\to 0}t=\lim\limits_{x\to 0}\dfrac{f(x)-1}{x}=\dfrac{1}{2}$。 + +\paragraph{关系转换} \leavevmode \medskip + +\textbf{例题:}如果$\lim\limits_{x\to 0}\dfrac{x-\sin x+f(x)}{x^4}$存在,则$\lim\limits_{x\to 0}\dfrac{x^3}{f(x)}$为常数多少? + +由$\lim\limits_{x\to 0}\dfrac{x\sin x+f(x)}{x^4}=A$,而目标是$x^3$,所以需要变形: + +$\lim\limits_{x\to 0}\dfrac{x\sin x+f(x)}{x^4}=A$ + +$\lim\limits_{x\to 0}\dfrac{x\sin x+f(x)\cdot x}{x^4}=A\cdot\lim\limits_{x\to 0}x=0$ + +$\lim\limits_{x\to 0}\dfrac{x-\sin x}{x^3}+\lim\limits_{x\to 0}\dfrac{f(x)}{x^3}=0$ + +$\text{泰勒展开:}x-\sin x=\dfrac{1}{6}x^3$ + +$\lim\limits_{x\to 0}\dfrac{f(x)}{x^3}=-\dfrac{1}{6}$ + +$\lim\limits_{x\to 0}\dfrac{x^3}{f(x)}=-6$ + +\paragraph{脱帽法} \leavevmode \medskip + +$\lim\limits_{x\to x_0}f(x)\Leftrightarrow f(x)=A+\alpha(x),\lim\limits_{x\to x_0}\alpha(x)=0$。 + +\textbf{例题:}如果$\lim\limits_{x\to 0}\dfrac{x-\sin x+f(x)}{x^4}$存在,则$\lim\limits_{x\to 0}\dfrac{x^3}{f(x)}$为常数多少? + +由$\lim\limits_{x\to 0}\dfrac{x\sin x+f(x)}{x^4}=A$脱帽:$\dfrac{x\sin x+f(x)}{x^4}=A+\alpha$。 + +得到:$f(x)=Ax^4+\alpha\cdot x^4-(x-\sin x)$。 + +反代入:$\lim\limits_{x\to 0}\dfrac{f(x)}{x^3}=\lim\limits_{x\to 0}\dfrac{Ax^4+\alpha\cdot x^4-x+\sin x}{x^3}=0+0-\dfrac{1}{6}=-\dfrac{1}{6}$。 + +$\therefore \lim\limits_{x\to 0}\dfrac{x^3}{f(x)}=-6$。 + +\subsubsection{求参数} + +因为求参数类型的题目中式子是未知的,所以求导后也是未知的,所以一般不要使用洛必达法则,而使用泰勒展开。 + +一般极限式子右侧等于一个常数,或是表明高阶或低阶。具体的关系参考无穷小比阶。 + +在求参数的时候要注意与0的关系。 + +\paragraph{常数} \leavevmode \medskip + +\textbf{例题:}设$\lim\limits_{x\to 0}\dfrac{\ln(1+x)-(ax+bx^2)}{x^2}=2$,求常数a,b。 + +根据泰勒展开式:$x\to 0,\ln(1+x)=x-\dfrac{x^2}{x}+o(x^2)$,$x-\ln(1+x)\sim\dfrac{1}{2}x^2\sim 1-\cos x$。 + +$\lim\limits_{x\to 0}\dfrac{\ln(1+x)-(ax+bx^2)}{x^2}=2$ + +$=\lim\limits_{x\to 0}\dfrac{(1-a)x-\left(\dfrac{1}{2}+b\right)x^2+o(x^2)}{x^2}=2\neq 0$ + +$1-a=0;-\left(\dfrac{1}{2}+b\right)=2$\medskip + +$\therefore a=1;b=-\dfrac{5}{2}$。 + +\paragraph{无穷小} \leavevmode \medskip + +若是求某个式子与另一个式子的某阶无穷小,则同右边等于常数一样,也需要使用泰勒展开。 + +\textbf{例题:}确定常数$a$和$b$,使得$f(x)=x-(a+b\cos x)\sin x$当$x\to 0$时关于$x$的5阶等价无穷小。 + +使用泰勒展开展开到五阶: + +$f(x)=x-(a+b\cos x)\sin x$ + +$=x-a\left[x-\dfrac{x^3}{3!}+\dfrac{x^5}{5!}+o(x^5)\right]-\dfrac{b}{2}\left[2x-\dfrac{(2x)^3}{3!}+\dfrac{(2x)^5}{5!}+o(x^5)\right]$ + +$=(1-a-b)x+\left(\dfrac{a}{6}+\dfrac{2b}{3}\right)x^3-\left(\dfrac{a}{120}+\dfrac{2b}{15}\right)x^5+o(x^5)$。\medskip + +所以若为五阶无穷小,则五阶前的常数应该都为0。 + +所以$1-a-b=0$,$\dfrac{a}{6}+\dfrac{2b}{3}=0$,$\dfrac{a}{120}+\dfrac{2b}{15}\neq0$。 + +解得$a=\dfrac{4}{3}$,$b=-\dfrac{1}{3}$。 + +\subsubsection{极限存在性} + +一般会给出带有参数的例子,并给定一个点指明在该点极限存在,求参数。 + +若该点极限存在,则该点两侧的极限都相等。\medskip + +\textbf{例题:}设函数$f(x)=\left\{\begin{array}{lcl} + \dfrac{\sin x(b\cos x-1)}{e^x+a}, & & x>0 \\ + \dfrac{\sin x}{\ln(1+3x)}, & & x<0 +\end{array} +\right.$在$x=0$处极限存在,则$a$,$b$分别为。 + +解:首先根据极限在$x=0$存在,且极限的唯一性。分段函数在0两侧的极限值必然相等。 + +$\because\lim\limits_{x\to 0^-}\dfrac{\sin x}{\ln(1+3x)}=\lim\limits_{x\to 0^-}\dfrac{\sin x}{3x}=\dfrac{1}{3}=\lim\limits_{x\to 0^+}\dfrac{\sin x(b\cos x-1)}{e^x+a}$。 + +\medskip + +又$\lim\limits_{x\to 0^+}\dfrac{\sin x(b\cos x-1)}{e^x+a}$的分母的$e^x$当$x\to 0^+$时$e^x\to 1$,假如$a\neq-1$,则$e^x+a\neq 0$,则为一个常数。 + +从而提取常数因子:$\lim\limits_{x\to 0^+}\dfrac{\sin x(b\cos x-1)}{e^x+a}=\dfrac{1}{1+a}\lim\limits_{x\to 0^+}\sin x(b\cos x-1)$,这时候$\sin x$是趋向0的,而$b\cos x-1$无论其中的$b$为何值都是趋向一个常数或0,这时候他们的乘积必然为无穷小,从而无法等于$\dfrac{1}{3}$这个常数。 + +$\therefore a=-1$,从而让极限式子变为一个商的形式:\medskip + +$\lim\limits_{x\to 0^+}\dfrac{\sin x(b\cos x-1)}{e^x+a}=\lim\limits_{x\to 0^+}\dfrac{\sin x(b\cos x-1)}{e^x-1}=\lim\limits_{x\to 0^+}\dfrac{\sin x(b\cos x-1)}{x}$\medskip + +$=\lim\limits_{x\to 0^+}b\cos x-1=b-1=\dfrac{1}{3}$\medskip + +$\therefore a=-1,b=\dfrac{4}{3}$。 + +\subsubsection{极限唯一性} + +若极限存在则必然唯一。 + +\textbf{例题:}设$a$为常数,$\lim\limits_{x\to 0}\left(\dfrac{e^{\frac{1}{x}}-\pi}{e^{\frac{2}{x}}+1}+a\cdot\arctan\dfrac{1}{x}\right)$存在,求出极限值。 + +因为求$x\to 0$,所以需要分两种情况讨论: + +\medskip + +$\lim\limits_{x\to 0^+}\left(\dfrac{e^{\frac{1}{x}}-\pi}{e^{\frac{2}{x}}+1}+a\cdot\arctan\dfrac{1}{x}\right)$ + +$=\lim\limits_{x\to 0^+}\left(\dfrac{e^{\frac{1}{x}}-\pi}{e^{\frac{2}{x}}+1}\right)+\lim\limits_{x\to 0^+}\left(a\cdot\arctan\dfrac{1}{x}\right)$ + +$=\lim\limits_{x\to 0^+}\left(\dfrac{0\cdot\left(e^{\frac{2}{x}}\right)^2+e^{\frac{1}{x}}-\pi}{1\cdot\left(e^{\frac{2}{x}}\right)^2+1}\right)+a\cdot\dfrac{\pi}{2}=a\cdot\dfrac{\pi}{2}$ + +\medskip + +$\lim\limits_{x\to 0^-}\left(\dfrac{e^{\frac{1}{x}}-\pi}{e^{\frac{2}{x}}+1}+a\cdot\arctan\dfrac{1}{x}\right)=-\pi+a\cdot\left(-\dfrac{\pi}{2}\right)=-\pi-\dfrac{\pi}{2}\cdot a$ + +因为极限值具有唯一性,所以$-\pi-\dfrac{\pi}{2}a=\dfrac{\pi}{2}a$,所以$a=-1$,极限值为$-\dfrac{\pi}{2}$。 + +\subsubsection{函数连续性} + +函数的连续性代表:极限值=函数值。所以函数的连续性需要靠极限完成。 + +\paragraph{极限判连续性} \leavevmode \medskip + +题目给出函数,往往是分段函数,然后判断分段点的连续性。\medskip + +\textbf{例题:}讨论函数$f(x)=\left\{\begin{array}{lcl} + \left[\dfrac{(1+x)^{\frac{1}{x}}}{e}\right]^{\frac{1}{x}},& & x>0 \\ + e^{-\frac{1}{2}}, & & x\leqslant 0 +\end{array}\right.$在$x=0$处的连续性。 + +因为$\lim\limits_{x\to 0^+}f(x)=\lim\limits_{x\to 0^+}\left[\dfrac{(1+x)^{\frac{1}{x}}}{e}\right]^{\frac{1}{x}}=e^{\lim\limits_{x\to 0^+}\frac{1}{x}\ln[\frac{(1+x)^{\frac{1}{x}}}{e}]}$。 + +又$\lim\limits_{x\to 0^+}\dfrac{1}{x}\ln\left[\dfrac{(1+x)^{\frac{1}{x}}}{e}\right]=\lim\limits_{x\to 0^+}\dfrac{1}{x}\left[\dfrac{1}{x}\ln(1+x)-1\right]=\lim\limits_{x\to 0^+}\dfrac{\ln(1+x)-x}{x^2}$ + +$=\lim\limits_{x\to 0^+}\dfrac{\dfrac{1}{1+x}-1}{2x}=\lim\limits_{x\to 0^+}-\dfrac{1}{2(1+x)}=-\dfrac{1}{2}$。 + +$\therefore\lim\limits_{x\to 0^+}f(x)=e^{-\frac{1}{2}}$。 + +又$\lim\limits_{x\to 0^-}f(x)=\lim\limits_{x\to 0^-}e^{-\frac{1}{2}}=e^{-\frac{1}{2}}$,且$f(0)=e^{-\frac{1}{2}}$。 + +从而$\lim\limits_{x\to 0^+}f(x)=\lim\limits_{x\to 0^-}f(x)=f(0)$,所以$f(x)$在$x=0$处连续。 + +\paragraph{连续性求极限} \leavevmode \medskip + +\textbf{例题:}函数在$f(x)$在$x=1$处连续,且$f(1)=1$,求$\lim\limits_{x\to+\infty}\ln\left[2+f\left(x^{\frac{1}{x}}\right)\right]$。 + +根据题目,所求的$\lim\limits_{x\to+\infty}\ln\left[2+f\left(x^{\frac{1}{x}}\right)\right]$中,唯一未知的且会随着$x\to+\infty$而变换就是$f\left(x^{\frac{1}{x}}\right)$。如果我们可以求出这个值就可以了。 + +而我们对于$f(x)$的具体的关系是未知的,只知道$f(1)=1$。那么先需要考察$\lim\limits_{x\to+\infty}x^{\frac{1}{x}}$的整数最大值。 + +$\lim\limits_{x\to+\infty}x^{\frac{1}{x}}=e^{\lim\limits_{x\to+\infty}\frac{\ln x}{x}}=e^{\lim\limits_{x\to+\infty}\frac{1}{x}}=e^0=1$ + +$\therefore\lim\limits_{x\to+\infty}f(x^{\frac{1}{x}})=f(1)=1$。 + +\subsubsection{迭代式数列} + +\paragraph{数列表达式} \leavevmode \medskip + +最重要的是将迭代式进行变形。 + +\textbf{例题:}数列$\{a_n\}$满足$a_0=0,a_1=1,2a_{n+1}=a_n+a_{n-1},n=1,2,\cdots$。计算$\lim\limits_{n\to\infty}a_n$。 + +首先看题目,给出的递推式设计到二阶递推,即存在三个数列变量,所以我们必须先求出对应的数列表达式。因为这个表达式涉及三个变量,所以尝试对其进行变型: + +$a_{n+1}-a_n=\dfrac{a_{n-1}-a_n}{2}=\left(-\dfrac{1}{2}\right)(a_n-a_{n-1})=\left(-\dfrac{1}{2}\right)^2(a_{n-1}-a_{n-2})$ + +$=\cdots$ + +$=\left(-\dfrac{1}{2}\right)^n(a_1-a_0)=\left(-\dfrac{1}{2}\right)^n$ + +然后得到了$a_{n+1}-a_n=\left(-\dfrac{1}{2}\right)^n$,而需要求极限,所以使用列项相消法的逆运算: + +$a_n=(a_n-a_{n-1})+(a_{n-1}-a_{n-2})+\cdots+(a_1-a_0)+a_0$\medskip + +$=\left(-\dfrac{1}{2}\right)^{n-1} + \left(-\dfrac{1}{2}\right)^{n-2} + \cdots + \left(-\dfrac{1}{2}\right)^0$\medskip + +$=\dfrac{1\cdot\left(1-\left(-\dfrac{1}{2}\right)^n\right)}{1-\left(-\dfrac{1}{2}\right)}=\dfrac{2}{3}\left[1-\left(-\dfrac{1}{2}\right)^n\right]$\medskip + +$\therefore\lim\limits_{n\to\infty}a_n=\dfrac{2}{3}$ + +\paragraph{单调有界准则} \leavevmode \medskip + +对于无法将关系式通过变形归纳为一般式的关系式,对于其极限就必须使用单调有界准则来求出。 + +单调有界的数列必有极限。需要证明单调性和有界性,然后对式子求极限就能求出目标极限。 + +\textbf{例题:}$x_0=0$,$x_n=\dfrac{1+2x_{n-1}}{1+x_{n-1}}(n\in N*)$,求$\lim\limits_{n\to\infty}x_n$。\medskip + +首先应该知道数列的趋向都是趋向正无穷。 + +然后对关系式进行变形:$x_n=\dfrac{1+2x_{n-1}}{1+x_{n-1}}=1+\dfrac{x_{n-1}}{1+x_{n-1}}=2-\dfrac{1}{1+x_{n-1}}$。 + +首先证明单调性,令$f(x)=2-\dfrac{1}{1+x}$。 + +$\therefore f'(x)=\dfrac{1}{(x+1)^2}>0$,则$f(x)$单调递增。 + +所以不管$x=x_{n-1}$或其他,$f'(x)>0$,$x_n$都是单调递增,则$x_n\geqslant x_0=0$。 + +然后证明有界性,$\because x_n\geqslant 0$且单调,$\therefore x_n=2-\dfrac{1}{1+x_{n-1}}\in[0,2]$。 + +从而$x_n$有界。 + +所以根据单调有界定理,$x_n$的极限存在。 + +对于关系式两边取极限: + +$\lim\limits_{n\to\infty}x_n=\lim\limits_{n\to\infty}\dfrac{1+2x_{n-1}}{1+x_{n-1}}=\dfrac{1+2\lim\limits_{n\to\infty}x_{n-1}}{1+\lim\limits_{n\to\infty}x_{n-1}}=\dfrac{1+2\lim\limits_{n\to\infty}x_n}{1+\lim\limits_{n\to\infty}x_n}$。 + +解该一元二次方程:$\lim\limits_{n\to\infty}x_n=\dfrac{1\pm\sqrt{5}}{2}$,又根据保号性,$\lim\limits_{n\to\infty}x_n>0$。 + +$\therefore\lim\limits_{n\to\infty}x_n=\dfrac{1+\sqrt{5}}{2}$。 + +而许多题目只给出样子,连通项公式都不会给出。\medskip + +\textbf{例题:}求出数列$\sqrt{2}$,$\sqrt{2+\sqrt{2}}$,$\sqrt{2+\sqrt{2+\sqrt{2}}}$$\cdots$的极限。 + +根据数列样式,无法通过普通的通项公式来表达,所以需要考虑使用递推式来表示:$x_{n+1}=\sqrt{2+x_n}$。 + +首先证明有界性: + +给定一个任意的正整数$k$,再根据递推式,假定$x_k<2$,所以$x_{k+1}=\sqrt{2+x_k}<\sqrt{2+2}=2$。且$x_1=\sqrt{2}$满足假定,所以$x_k<2$对于任意的正整数$k$都成立,所以$x_n$存在上界2。 + +然后证明单调性,根据其递推式: + +$x_{n+1}-x_n=\sqrt{2+x_n}-x_n=\dfrac{2+x_n-x_n^2}{\sqrt{2+x_n}+x_n}=\dfrac{-(x_n-2)(x_n+1)}{\sqrt{2+x_n}+x_n}$。\medskip + +又$0Z2vu@RW*t^boSa1E_HAdApx(B(E zm;^mD13L`)(cSeM3@Zl_6On_lH4GmgqlA^MtC=&SgsqXQnV6Z0gQ*##oSD6at0fUL z6B~FUX-K?AX4CE4_)si-@()FVoF@q5jFLV@F1;wO~BkN5ZV%`1H4V>R3EudxYGSHZft zV)-YqC75C3rRgeg|7K?K_9&3G3I55e;_T<~^E9j#6PL(7Cu5NbiI2Bf#!kKrD6`B} zy7syS6#PaU31tBjk7mWj;@6;i#Lc+y&MbzK)x| zv-975KGiNaMs3Lp)6joCqYB!>{4YZo4y-vI-OsD4Jv>mhte}}Q-uEl!qg87M&v-EL zs#@I?GHWeD1YjiPZYdl?6>ke={}>r_7re-TM$hIg*{5t-l;?7oc7>FIB}piW!0|9k z2>X+K*I*^h9|I4y9gmE5fFs)Ef`VkZHq^NS{TjQCxP!v6QJ`vGA5rN2D}luOlhM;( 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+\usepackage{color} +% 使用颜色 +\definecolor{orange}{RGB}{255,127,0} +\usepackage{geometry} +\setcounter{tocdepth}{4} +\setcounter{secnumdepth}{4} +% 设置四级目录与标题 +\geometry{papersize={21cm,29.7cm}} +% 默认大小为A4 +\geometry{left=3.18cm,right=3.18cm,top=2.54cm,bottom=2.54cm} +% 默认页边距为1英尺与1.25英尺 +\usepackage{indentfirst} +\setlength{\parindent}{2.45em} +% 首行缩进2个中文字符 +\usepackage{setspace} +\renewcommand{\baselinestretch}{1.5} +% 1.5倍行距 +\usepackage{amssymb} +% 因为所以 +\usepackage{amsmath} +% 数学公式 +\usepackage[colorlinks,linkcolor=black,urlcolor=blue]{hyperref} +% 超链接 +\author{Didnelpsun} +\title{不定积分与定积分} +\date{} +\begin{document} +\maketitle +\pagestyle{empty} +\thispagestyle{empty} +\tableofcontents +\thispagestyle{empty} +\newpage +\pagestyle{plain} +\setcounter{page}{1} +\section{不定积分} +\subsection{基本积分} + +\textbf{例题:}汽车以20m/s的速度行驶,刹车后匀减速行驶了50m停止,求刹车加速度。 + +已知题目含有两个变量:距离和时间,设距离为$s$,时间为$t$。 + +因为汽车首先按20m/s匀速运动,所以$\dfrac{\textrm{d}s}{\textrm{d}t}\bigg\vert_{t=0}=20$,最开始距离为0,所以$s\vert_{t=0}=0$。 + +又因为是匀减速的,所以速度形如:$v=\dfrac{s}{t}=kt+b$,从而令二阶导数下$\dfrac{\textrm{d}^2s}{\textrm{d}t^2}=k$。 + +所以$\displaystyle{\dfrac{\textrm{d}s}{\textrm{d}t}=\int\dfrac{\textrm{d}^2s}{\textrm{d}t^2}\,\textrm{d}t=\int k\,\textrm{d}t}=kt+C_1$。 + +代入$\dfrac{\textrm{d}s}{\textrm{d}t}\bigg\vert_{t=0}=20$,所以$C_1=20$,即$\dfrac{\textrm{d}s}{\textrm{d}t}=kt+20$。 + +所以$\textrm{d}s=(kt+20)\,\textrm{d}t$,从而$s=\displaystyle{\int(kt+20)\,\textrm{d}t}=\dfrac{1}{2}kt^2+20t+C_2$。 + +又$s\vert_{t=0}=0$,所以代入得$C_2=0$,所以$s=\dfrac{1}{2}kt^2+20t$。 + +当$s=50$时停住,所以此时$\dfrac{\textrm{d}s}{\textrm{d}t}=0$,得到$t=-\dfrac{20}{k}$。 + +代入$s$:$50=\dfrac{1}{2}k\left(-\dfrac{20}{k}\right)^2+20\left(-\dfrac{20}{k}\right)$,解得$k=-4$,即加速度为-4m/$s^2$。 + +\subsection{换元积分} + +\subsubsection{第一类换元} + +\paragraph{聚集因式} \leavevmode \medskip + +将复杂的式子转换为简单的一个因式放到$\textrm{d}$后面看作一个整体,然后利用基本积分公式计算。 + +\textbf{例题:}$\displaystyle{\int\dfrac{\textrm{d}x}{x\ln x\ln\ln x}}$。 \medskip + +$=\displaystyle{\int\dfrac{\textrm{d}(\ln x)}{\ln x\ln\ln x}}=\displaystyle{\int\dfrac{\textrm{d}(\ln\ln x)}{\ln\ln x}}=\ln\vert\ln\ln x\vert+C$。\medskip + +\textbf{例题:}$\displaystyle{\int\dfrac{10^{2\arccos x}}{\sqrt{1-x^2}}\,\textrm{d}x}$。 + +$=-\displaystyle{\int10^{2\arccos x}\,\textrm{d}(\arccos x)}=-\dfrac{1}{2}\displaystyle{\int10^{2\arccos x}\,\textrm{d}(2\arccos x)}=-\dfrac{10^{2\arccos x}}{2\ln10}+C$。 + +\paragraph{积化和差} \leavevmode \medskip + +对于两个三角函数的乘积可以使用积化和差简单计算。 + +\textbf{例题:}$\displaystyle{\int\sin2x\cos3x\,\textrm{d}x}$。 + +$=\displaystyle{\int\cos3x\sin2x\,\textrm{d}x=\dfrac{1}{2}\int(\sin5x-\sin x)\,\textrm{d}x}$ + +$=\dfrac{1}{2}\int\sin5x\,\textrm{d}x-\dfrac{1}{2}\int\sin x\,\textrm{d}x=-\dfrac{1}{10}\cos5x+\dfrac{1}{2}\cos x+C$。 + +\paragraph{三角拆分} \leavevmode \medskip + +主要用于$\sec^2-1=\tan^2x$,当出现$\tan^2$、$\tan^3$等与$\sec x$在一起作为乘积时可以考虑拆分。 + +\textbf{例题:}$\displaystyle{\int\tan^3x\sec x\,\textrm{d}x}$。 + +$=\displaystyle{\int(\sec^2x-1)}\tan x\sec x\,\textrm{d}x=\displaystyle{\int(\sec^2x-1)}\,\textrm{d}(\sec x)=\dfrac{1}{3}\sec^3x-\sec x+C$。 + +\subsubsection{第二类换元} + +\begin{enumerate} + \item $\sqrt{a^2-x^2}$:$x=a\sin t(a\cos t)$。 + \item $\sqrt{a^2+x^2}$:$x=a\tan t$。 + \item $\sqrt{x^2-a^2}$:$x=a\sec t$。 +\end{enumerate} + +\subsection{分部积分} + +\subsubsection{基本分部} + +\subsubsection{多次分部} + +\subsection{有理积分} + +\subsubsection{高阶多项式分配} + +当不定积分式子形如$\displaystyle{\int\dfrac{f(x)}{g(x)}\,\textrm{d}x}$,且$f(x)$、$g(x)$都为与$x$相关的多项式,$f(x)$阶数高于或等于$g(x)$,则$f(x)$可以按照$g(x)$的形式分配,约去式子,得到最简单的表达。 + +\textbf{例题:}$\displaystyle{\int\dfrac{x^3}{x^2+9}\,\textrm{d}x}$。 \medskip + +$=\displaystyle{\int\dfrac{x^3+9x-9x}{x^2+9}\,\textrm{d}x=\int\dfrac{x^3+9x}{x^2+9}\,\textrm{d}x-\int\dfrac{9x}{x^2+9}\,\textrm{d}x}$ \medskip + +$\displaystyle{=\int x\,\textrm{d}x-\dfrac{9}{2}\int\dfrac{\textrm{d}(x^2+9)}{x^2+9}}=\dfrac{x^2}{2}-\dfrac{9}{2}\ln(9+x^2)+C$。 + +\subsubsection{低阶多项式分解} + +当不定积分式子形如$\displaystyle{\int\dfrac{f(x)}{g(x)}\,\textrm{d}x}$,且$f(x)$、$g(x)$都为与$x$相关的多项式,$f(x)$阶数低于$g(x)$,则可以分解式子:$\displaystyle{\int\dfrac{f(x)}{g(x)}\,\textrm{d}x=\int\dfrac{f_1(x)}{g_1(x)}\,\textrm{d}x+\int\dfrac{f_2(x)}{g_2(x)}\,\textrm{d}x}$。 + + + +\section{定积分} + +\subsection{变限积分} + +\subsection{牛莱公式} + +\subsection{换元积分} + +\subsection{分部积分} + +\subsection{反常积分} + +\section{积分应用} + +\subsection{面积} + +\subsection{体积} + +\subsection{弧长} + +\end{document} diff --git a/advanced-math/knowledge/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.pdf b/advanced-math/knowledge/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.pdf index 55f56efee6d58af33fa3e3382b55c3cabad01da0..dedba8c2ff2f343df268545538d4b89fb836f3d7 100644 GIT binary patch delta 127503 zcmZU)V~{4mvIW?lwr$(CZQHi3uWh?~+O}=m?rGaLXXn26HsWsVpNfjA%=%GvDkD$g zHVJw%5xPDE41|>{5s4JQ#>~ji5AEXWY-VH!?fK_gbEjum*u$*}8%o_tI1x;;b^fT?hT`=vW=q1q^I*&aRk7AtdG7-* 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a/advanced-math/knowledge/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.tex b/advanced-math/knowledge/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.tex index 66d75b5..6a86083 100644 --- a/advanced-math/knowledge/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.tex +++ b/advanced-math/knowledge/4-indefinite-integral-and-definite-integral/indefinite-integral-and-definite-integral.tex @@ -402,11 +402,11 @@ $\int_a^bf(x)\,\textrm{d}x=F(b)-F(a)=F'(\xi)(b-a)=f(\xi)(b-a)(a<\xi b)$。 元素法也叫微元法,是分析、解决物理问题中的常用方法,也是从部分到整体的思维方法。用该方法可以使一些复杂的物理过程用我们熟悉的物理规律迅速地加以解决,使所求的问题简单化。在使用元素法处理问题时,需将其分解为众多微小的“元过程”,而且每个“元过程”所遵循的规律是相同的,这样,我们只需分析这些“元过程”,然后再将“元过程”进行必要的数学方法或物理思想处理,进而使问题求解。 -\subsection{几何应用} +数一基本上只考几何应用不考物理应用。 -\subsubsection{面积} +\subsection{面积} -\paragraph{直角坐标系} \leavevmode \medskip +\subsubsection{直角坐标系} \textbf{例题:}求曲线$y^2=x$与$y=x^2$所围成面积。 @@ -430,7 +430,7 @@ $\textrm{d}S=\left[(y+4)-\dfrac{y^2}{2}\right]\textrm{d}y$。 所以$S=\displaystyle{\int_{-2}^4\left[(y+4)-\dfrac{y^2}{2}\right]\textrm{d}y}$ -\paragraph{参数方程} \leavevmode \medskip +\subsubsection{参数方程} \textbf{例题:}求摆线一拱$\left\{\begin{array}{l} x=a(t-\sin t) \\ @@ -456,7 +456,7 @@ $=8a^2\int_0^\pi\sin^4u\,\textrm{d}u=16a^2\int_0^{\frac{\pi}{2}}\sin^4u\,\textrm $=16a^2\cdot\dfrac{3}{4}\cdot\dfrac{1}{2}\cdot\dfrac{\pi}{2}=3a^2\pi$(点火公式)。 -\paragraph{极坐标} \leavevmode \medskip +\subsubsection{极坐标} 已知极径函数$\rho=\rho(\theta)$,极角$\theta\in[\alpha,\beta]$,极坐标所围成面积就是初始角所在射线与结束角所在射线以及函数所围成的图形。所以微元计算时所围成的图形可以近似看作扇形。 @@ -476,9 +476,9 @@ $=a^2\displaystyle{\int_0^\pi\left(2\cos^2\dfrac{\theta}{2}\right)^2\textrm{d}\t 根据华理士公式:$=8a^2\cdot\dfrac{3}{4}\cdot\dfrac{1}{2}\cdot\dfrac{\pi}{2}=\dfrac{3}{2}a^2\pi$。 -\subsubsection{体积} +\subsection{体积} -\paragraph{旋转体} \leavevmode \medskip +\subsubsection{旋转体} 当绕$x$轴进行旋转,可以看作从$x$轴沿$y$轴水平切割旋转体,就得到了以$x$轴为中心的一个圆柱,底边半径为$f(x)$,高度为$\textrm{d}x$,所以$\textrm{d}V_x=\pi f^2(x)\,\textrm{d}x$,所以$V_x=\pi\int_a^bf^2(x)\,\textrm{d}x$(如果用$y(x)$表达,就是$V_x=\pi\int_c^d\varphi^2(y)\,\textrm{d}y$)。 @@ -520,7 +520,7 @@ $=2\pi\int_0^{2\pi}a(t-\sin t)a^2(1-\cos t)^2\,\textrm{d}t=2a^3\pi\int_0^{2\pi}( 然后拆开分别进行凑微分法,得到$6a^3\pi^3$。 -\paragraph{平行截面已知的立体体积} \leavevmode \medskip +\subsubsection{平行截面已知的立体体积} 已知截面面积可以通过对应的高得到立体体积:$V=\int_a^bS(x)\,\textrm{d}x$。 @@ -534,7 +534,7 @@ $V=2\int_0^a\pi bc\left(1-\dfrac{x^2}{a^2}\right)\,\textrm{d}x$。 解得$V=\dfrac{4}{3}\pi abc$。 -\subsubsection{弧长} +\subsection{弧长} 在弧长中插入$n$个点$M_1,M_2,\cdots,M_{i-1},M_i,\cdots,M_n$。