Evaluate the following limit: _ x 2 ^2x 1- - Vidyadip

Question

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

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Answer

$\lim\limits_{\text{x}\rightarrow\frac{\pi}{2}}\frac{\cos^2\text{x}}{1-\sin\text{x}}$
$=\lim\limits_{\text{x}\rightarrow\frac{\pi}{2}}\frac{1-\sin^2\text{x}}{1-\sin\text{x}}$
$=\lim\limits_{\text{x}\rightarrow\frac{\pi}{2}}\frac{(1-\sin\text{x})(1+\sin\text{x})}{(1-\sin\text{x})}$
$=\lim\limits_{\text{x}\rightarrow\frac{\pi}{2}}{(1+\sin\text{x})}$
$=1+\sin\frac{\pi}{2}$
$=1+1$
$=2$

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