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

Question

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

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Answer

$\lim\limits_{\text{x}\rightarrow{\frac{\pi}{2}}}\frac{\sqrt{2}-\sqrt{1+\sin\text{x}}}{\cos^2\text{x}}$$=\lim\limits_{\text{x}\rightarrow{\frac{\pi}{2}}}\frac{\sqrt{2}-\sqrt{1+\sin\text{x}}}{\cos^2\text{x}}\frac{\sqrt{2}+\sqrt{1+\sin\text{x}}}{\sqrt{2}+\sqrt{1+\sin\text{x}}}$
$=\lim\limits_{\text{x}\rightarrow{\frac{\pi}{2}}}\frac{2-1-\sin\text{x}}{\cos^2\text{x}\big(\sqrt{2}-\sqrt{1+\sin\text{x}}\big)}$
$=\lim\limits_{\text{x}\rightarrow{\frac{\pi}{2}}}\frac{1-\sin\text{x}}{\big(1-\sin^2\text{x}\big)\big(\sqrt{2}-\sqrt{1+\sin\text{x}}\big)}$
$=\lim\limits_{\text{x}\rightarrow{\frac{\pi}{2}}}\frac{1}{\big(1+\sin\text{x}\big)\big(\sqrt{2}+\sqrt{1+\sin\text{x}}\big)}$
$=\frac{1}{(1+1)\big(\sqrt{2}+\sqrt{2}\big)}$
$=\frac{1}{\big(4\sqrt{2}\big)}$

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