Question

The value of $f(0)$, such that $f(x)=\frac{1}{x^{2}}(1-\cos (\sin x))$ can be made continuous at $x=0,$ is

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

Answer 1

Answer: (A)

$\displaystyle\lim _{x \rightarrow 0} \frac{1-\cos (\sin x)}{x^{2}}=\displaystyle\lim _{x \rightarrow 0} \frac{2 \sin ^{2}\left(\frac{\sin x}{2}\right)}{x^{2}} \displaystyle\lim _{x \rightarrow 0} 2 \cdot\left(\frac{\sin \left(\frac{\sin x}{2}\right)}{\left(\frac{\sin x}{2}\right)}\right)^{2} \cdot\left(\frac{\sin x}{2 x}\right)^{2}=\frac{1}{2}$
Thus, for continuity at $x=0,$
$f\left(\right.0\left.\right)=\frac{1}{2}$