Question

If $f:R\to R$ is defined by
$f(x)=\{\begin{array}{ll}\frac{2\sin x-\sin 2x}{2x\cos x},& \text{ if }x\ne 0\\ a,& \text{ if }x=0\end{array}$
then the value of a so that f is continuous at $0$ is

• A. 2
• B. 1
• C. -1
• D. 0

Answer 1

Answer: (D)

Given, $f(x)=\{\begin{array}{ll}\frac{2\sin x-\sin 2x}{2x\cos x},& \text{ if }x\ne 0\\ a,& \text{ if }x=0\end{array}$
= $\underset{x\to 0}{\lim }f(x)=\underset{x\to 0}{\lim }\frac{2\sin x-\sin 2x}{2x\cos x}(\frac{0}{0}$ form $)$
= $\underset{x\to 0}{\lim }\frac{2\cos x-2\cos 2x}{2\left(\cos x-x\sin x\right)}$
= $\underset{x\to 0}{\lim }\frac{2-2}{2\left(1-0\right)}=0$
Since, $f(x)$ is continuous at $x=0$
$\therefore f(0)=\underset{x\to 0}{\lim }f(x)$
$\Rightarrow a=0$