Evaluate the following limit: _ x1 1+ (1-x)^2 - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow{1}}\frac{1+\cos\pi\text{x}}{(1-\text{x})^2}$

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Answer

$\lim\limits_{\text{x}\rightarrow{1}}\frac{1+\cos\pi\text{x}}{(1-\text{x})^2}$
$\Rightarrow\text{x}\rightarrow1,\text{x}-1\rightarrow0,$ let $\text{x}-1=\text{y}$
$=\lim\limits_{\text{y}\rightarrow{0}}\frac{1+\cos\pi(\text{y}+1)}{(-\text{y})^2}$
$=\lim\limits_{\text{y}\rightarrow{0}}\frac{1+\cos(\pi+\pi\text{y})}{\text{y}^2}$
$=\lim\limits_{\text{y}\rightarrow{0}}\frac{1-\cos(\pi\text{y})}{\text{y}^2}$
$=\lim\limits_{\text{y}\rightarrow{0}}\frac{2-\sin^2\frac{\pi\text{y}}{2}}{\text{y}^2}$
$=2\Bigg(\lim\limits_{\text{y}\rightarrow{0}}\frac{\sin\frac{\pi\text{y}}{2}}{\frac{\pi\text{y}}{2}}\Bigg)^2\times\frac{\pi^2}{4}$
$=2\times1\times\frac{\pi^2}{4}$
$=\frac{\pi^2}{2}$

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