Question

A function $f\left(x\right)$ is defined as below $f\left(x\right)=\frac{\cos \left(\sin x\right)-\cos x}{x^2},x\ne 0$ and $f\left(0\right)=a$. $f\left(x\right)$ is continuous at $x=0$ if $a$ equals

• A. $0$
• B. $4$
• C. $5$
• D. $6$

Answer 1

The correct option is A $0$

By L'Hospital's Rule,

$=\underset{x\to 0}{lim}\frac{-\sin \left(\sin x\right).\cos x+\sin x}{2x}$

$=\underset{x\to 0}{lim}\frac{\sin \left(\sin x\right)\sin x-{\cos }^{2}x.\cos \left(\sin x\right)+\cos x}{2}$

$=\frac{-{\cos }^{2}0.\cos \left(0\right)+\cos 0}{2}$

$=\frac{0}{2}$

$=0=a$