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

Question

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

✓

Answer

$\lim\limits_{\text{x}\rightarrow{\frac{\pi}{2}}}\Big(\frac{\pi}{2}-\text{x}\Big)\tan\text{x}$
Let $\text{y} =\frac{\pi}{2}-\text{x}$ as $\text{x}\rightarrow\frac{\pi}{2},\text{y}\rightarrow0$
$=\lim\limits_{\text{x}\rightarrow\frac{\pi}{2}}\Big(\frac{\pi}{2}-\text{x}\Big)\tan\text{x}$
$=\lim\limits_{\text{y}\rightarrow0}\text{ y }\tan\Big(\frac{\pi}{2}-\text{y}\Big)$
$=\lim\limits_{\text{y}\rightarrow0}\text{ y }\frac{\sin\big(\frac{\pi}{2}-\text{y}\big)}{\cos\big(\frac{\pi}{2}-\text{y}\big)}$
$=\lim\limits_{\text{y}\rightarrow0}\text{ y }\frac{\cos\text{y}}{\sin\text{y}}$
$=\lim\limits_{\text{y}\rightarrow0}\cos\text{ y }\lim\limits_{\text{y}\rightarrow0}\frac{\text{y}}{\sin\text{y}}$
$=1$

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