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

Question

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

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Answer

$\lim\limits_{\text{x}\rightarrow{\frac{\pi}{2}}}\frac{1-\sin\text{x}}{\big(\frac{\pi}{2}-\text{x}\big)}$
If $\text{x}\rightarrow\frac{\pi}{2},\frac{\pi}{2}-\text{x}\rightarrow0$
Let $\frac\pi2-\text{x}=\text{y}$ they y → 0
$=\lim\limits_{\text{y}\rightarrow{0}}\frac{1-\sin\big(\frac{\pi}{4}-\text{y}\big)}{\text{y}^2}$
$=\lim\limits_{\text{y}\rightarrow{0}}\frac{1-\cos\text{y}}{\text{y}^2}$
$=\lim\limits_{\text{y}\rightarrow{0}}\frac{2\sin^2\frac{\text{y}}{2}}{\text{y}^2}$
$=2\Bigg(\lim\limits_{\text{y}\rightarrow{0}}\frac{\sin\frac{\text{y}}{2}}{\frac{\text{y}}{2}}\Bigg)^2\times\frac14$ $\Big[\because\lim\limits_{\text{x}\rightarrow{0}}\frac{\sin\text{x}}{\text{x}}=1\Big]$
$=2\times1\times\frac14$
$=\frac12$

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