Question

Solve : $\underset{x\to \pi /4}{lim}\frac{\sqrt{2}\cos x-1}{\cot x-1}$

Answer 1

$\underset{x\to \pi /4}{lim}\frac{\sqrt{2}\cos x-1}{\cot x-1}=\frac{0}{0}$ form

Using $L^{\prime }Hospita{l}^{\prime }s$ Rule

$\underset{x\to \pi /4}{lim}\frac{\sqrt{2}\sin x}{1/{\sin }^{2}x}$

$\underset{x\to \pi /4}{lim}\sqrt{2}{\sin }^{3}x$

$=\sqrt{2}{\left(\sin \left(\frac{\pi }{4}\right)\right)}^3$

$=\sqrt{2}.\frac{1}{(\sqrt{2}{)}^3}$

$=\frac{1}{2}$