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# 関数の極限の求め方

$3\displaystyle{ \infty- \infty }$$3\displaystyle{\frac{\hspace{2}0\hspace{2}}{\hspace{2}0\hspace{2}}}$$3\displaystyle{\frac{\hspace{2}\infty\hspace{2}}{\hspace{2}\infty\hspace{2}}}$極限が求められる形に式を変形する．

## ■変形方法

### ● 因数分解→約分

$3\displaystyle{\begin{array}{lll} { \lim }\limits_{ x\rightarrow 1 }\frac{\hspace{2} {x}^{2}+x- 2 \hspace{2}}{\hspace{2} {x}^{2}+2x- 3 \hspace{2}} & ={ \lim }\limits_{ x\rightarrow 1 }\frac{\hspace{2} \( x- 1 \) \( x+2 \) \hspace{2}}{\hspace{2} \( x- 1 \) \( x+3 \) \hspace{2}} & \vspace{6}\\ & ={ \lim }\limits_{ x\rightarrow 1 }\frac{\hspace{2} x+2 \hspace{2}}{\hspace{2} x+3 \hspace{2}} & \vspace{6}\\ & =\frac{\hspace{2}3\hspace{2}}{\hspace{2}4\hspace{2}} & \vspace{6}\\ \end{array}}$

### ● $3\displaystyle{{ \lim }\limits_{ x\rightarrow \infty }\frac{\hspace{2}1\hspace{2}}{\hspace{2} {x}^{n} \hspace{2}}}$を作り出す

#### ・最高次数でくくりだし

$3\displaystyle{\begin{array}{lll} { \lim }\limits_{ x\rightarrow \infty } \( 3{x}^{3}+2{x}^{2}+5 \) & ={ \lim }\limits_{ x\rightarrow \infty }{x}^{3} \( 3- \frac{\hspace{2}2\hspace{2}}{\hspace{2}x\hspace{2}}- \frac{\hspace{2}5\hspace{2}}{\hspace{2} {x}^{3} \hspace{2}} \) & \vspace{6}\\ & =\infty\times \( 3- 2\times 0- 5\times 0 \) & \vspace{6}\\ & =\infty\times 3 & \vspace{6}\\ & =\infty & \vspace{6}\\ \end{array}}$

#### ・$3\displaystyle{{x}^{n}}$  で割る

$3\displaystyle{\begin{array}{lll} { \lim }\limits_{ x\rightarrow \infty }\frac{\hspace{2} {x}^{2}+3x+4 \hspace{2}}{\hspace{2} 2{x}^{2}+1 \hspace{2}} & ={ \lim }\limits_{ x\rightarrow \infty }\frac{\hspace{2} 1+\frac{\hspace{2}3\hspace{2}}{\hspace{2}x\hspace{2}}+\frac{\hspace{2}4\hspace{2}}{\hspace{2} {x}^{2} \hspace{2}} \hspace{2}}{\hspace{2} 2+\frac{\hspace{2}1\hspace{2}}{\hspace{2} {x}^{2} \hspace{2}} \hspace{2}} & \vspace{6}\\ & =\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} & \vspace{6}\\ \end{array}}$

### ● 有理化

$3\displaystyle{\begin{array}{lll} { \lim }\limits_{ x\rightarrow 2 }\frac{\hspace{2} \sqrt{ x+2 }- 2 \hspace{2}}{\hspace{2} x- 2 \hspace{2}} & ={ \lim }\limits_{ x\rightarrow 2 }\frac{\hspace{2} \( \sqrt{ x+2 }- 2 \) \( \sqrt{ x+2 }+2 \) \hspace{2}}{\hspace{2} \( x- 2 \) \( \sqrt{ x+2 }+2 \) \hspace{2}} & \vspace{6}\\ & ={ \lim }\limits_{ x\rightarrow 2 }\frac{\hspace{2} x+2- {2}^{2} \hspace{2}}{\hspace{2} \( x- 2 \) \( \sqrt{ x+2 }+2 \) \hspace{2}} & \vspace{6}\\ & ={ \lim }\limits_{ x\rightarrow 2 }\frac{\hspace{2} x- 2 \hspace{2}}{\hspace{2} \( x- 2 \) \( \sqrt{ x+2 }+2 \) \hspace{2}} & \vspace{6}\\ & ={ \lim }\limits_{ x\rightarrow 2 }\frac{\hspace{2}1\hspace{2}}{\hspace{2} \sqrt{ x+2 }+2 \hspace{2}} & \vspace{6}\\ & =\frac{\hspace{2}1\hspace{2}}{\hspace{2}4\hspace{2}} & \vspace{6}\\ \end{array}}$

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