$\lim\limits_{x\to \large\frac{\pi}{4}}\large\frac{\sqrt 2\cos x-1}{\cos x-1}$=

Question

$(a)\;\large\frac{1}{\sqrt 2}$$\qquad(b)\;\large\frac{1}{2}$$\qquad(c)\;\large\frac{1}{2\sqrt 2}\qquad$$(d)\;1$

Answer 1

$\lim\limits_{x\to \large\frac{\pi}{4}}\large\frac{\sqrt 2\cos x-1}{\cot x-1}\qquad$$(0/0)$ Applying L Hospital's Rule

$\Rightarrow \lim\limits_{x\to\Large\frac{\pi}{4}}\large\frac{-\sqrt 2\sin x}{-cosec^2x}$

$\Rightarrow \sqrt 2\lim\limits_{x\to \large\frac{\pi}{4}}\sin^3x$

$\Rightarrow \sqrt 2\big(\large\frac{1}{\sqrt 2}\big)^3$

$\Rightarrow \large\frac{1}{2}$

Hence (b) is the correct answer.