_ x / 4 2 x-1 x-1 = - Vidyadip

MCQ

$\lim _{x \rightarrow \pi / 4} \frac{\sqrt{2} \cos x-1}{\cot x-1}=$

A
$\frac{1}{\sqrt{2}}$
✓
$\frac{1}{2}$
C
$\frac{1}{2 \sqrt{2}}$
D
1

✓

Answer

Correct option: B.

$\frac{1}{2}$

(B)
Applying L-Hospital's rule, we get
$\lim _{x \rightarrow \frac{\pi}{4}} \frac{\sqrt{2} \cos x-1}{\cot x-1}=\lim _{x \rightarrow \frac{\pi}{4}} \frac{-\sqrt{2} \sin x}{-\operatorname{cosec}^2 x}$
$=\frac{\sqrt{2} \times \frac{1}{\sqrt{2}}}{(\sqrt{2})^2}=\frac{1}{2}$

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