Question

If $f: R \rightarrow R$ is defined by
$f(x)=\begin{cases}\frac{2 \sin x-\sin 2 x}{2 x \cos x}, & \text { if } x \neq 0 \\ a, & \text { if } x=0\end{cases}$
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, $f(x)=\begin{cases}\frac{2 \sin x-\sin 2 x}{2 x \cos x}, & \text { if } x \neq 0 \\ a, & \text { if } x=0\end{cases}$
= $\displaystyle \lim_{x \to 0}f(x)= \displaystyle \lim_{x \to 0}\frac{2 \sin x-\sin 2x }{2x \cos x}(\frac{0}{0}$ form $)$
= $\displaystyle \lim_{x \to 0}\frac{2 \cos x -2 \cos 2x}{2\left(\cos x -x \sin x\right)}$
= $\displaystyle \lim_{x \to 0}\frac{2-2}{2\left(1-0\right)}=0$
Since, $f(x)$ is continuous at $x=0$
$\therefore f(0)=\displaystyle\lim _{x \rightarrow 0} f(x) $
$\Rightarrow a=0$