_ 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
B
$0$
C
-1
D
does not exit

✓

Answer

(b) $0$
Explanation: $\lim _{x \rightarrow \frac{\pi}{2}} \frac{1-\sin x}{\cos x}=\lim _{y \rightarrow 0} \frac{1-\sin \left(\frac{\pi}{2}-y\right)}{\cos \left(\frac{\pi}{2}-y\right)} \quad$ taking $\frac{\pi}{2}-x=y$
$\begin{array}{l}=\lim _{y \rightarrow 0} \frac{1-\cos y}{\sin y}=\lim _{y \rightarrow 0} \frac{2 \sin ^2 \frac{y}{2}}{2 \sin \frac{y}{2} \cos \frac{y}{2}} \\ =\lim _{y \rightarrow 0} \tan \frac{y}{2}=0\end{array}$

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