Question

solve.
$\underset{x\to \frac{\pi }{4}}{lim}\frac{\sqrt{2}-\cos x-\sin x}{{\left(\frac{\pi }{4}-x\right)}^2}$

• A. $\frac{4}{\sqrt{2}}$
• B. $\frac{2}{\sqrt{2}}$
• C. $\frac{-1}{\sqrt{2}}$
• D. $\frac{1}{\sqrt{2}}$

Answer 1

Answer: (C)

$\frac{-1}{\sqrt{2}}$

$\underset{x\to \frac{\pi }{4}}{lim}\frac{\sqrt{2}-\cos x-\sin x}{{(\frac{\pi }{4}-x)}^2}$

Apply L hospital rule two times

$=\frac{\sin x-\cos x}{2(\frac{\pi }{4}-x)}$.

$=\frac{\cos x+\sin x}{-2}$ at $x\to \frac{\pi }{4}$

$=\frac{\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}}{-2}$

$=\frac{-1}{\sqrt{2}}$