_ x 2 1- x x is equal to - Vidyadip

MCQ

$\lim _{x \rightarrow \frac{\pi}{2}} \frac{1-\sin x}{\cos x}$ is equal to

A
$1$
✓
$0$
C
$-1$
D
does not exit

✓

Answer

Correct option: B.

$0$

$\lim _{x \rightarrow \frac{\pi}{2}} \frac{1-\sin x}{\cos x}$
$=\lim _{y \rightarrow 0} \frac{1-\sin \left(\frac{\pi}{2}-y\right)}{\cos \left(\frac{\pi}{2}-y\right)}$ taking $\frac{\pi}{2}-x=y$
$=\lim _{y \rightarrow 0} \frac{1-\cos y}{\sin y}$
$=\lim _{y \rightarrow 0} \frac{2 \sin ^2 \frac{y}{2}}{2 \sin \frac{y}{2} \cos \frac{y}{2}}$
$=\lim _{y \rightarrow 0} \tan \frac{y}{2}=0$

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