If $f: R \rightarrow R$ is defined by $f(x)=\left\{\begin{array}{cc} \frac{2 \sin x-\sin 2 x}{2 x \cos x}, &…
If $f: R \rightarrow R$ is defined by
$f(x)=\left\{\begin{array}{cc}
\frac{2 \sin x-\sin 2 x}{2 x \cos x}, & \text { if } x \neq 0 \\
a, & \text { if } x=0
\end{array}\right. \text {, }$
then the value of $a$ so that $f$ is continuous at 0 is
2
1
-1
0
Solution
Given, $f(x)=\left\{\begin{array}{cc}\frac{2 \sin x-\sin 2 x}{2 x \cos x}, & \text { if } x \neq 0 \\ a, & \text { if } x=0\end{array}\right.$
Now, $\lim _{x \rightarrow 0} f(x)=\lim _{x \rightarrow 0} \frac{2 \sin x-\sin 2 x}{2 x \cos x}$
$\begin{aligned}
& \quad\left(\frac{0}{0} \text { form }\right) \\
& =\lim _{x \rightarrow 0} \frac{2 \cos x-2 \cos 2 x}{2(\cos x-x \sin x)} \\
& =\lim _{x \rightarrow 0} \frac{2-2}{2(1-0)}=0
\end{aligned}$
Since, $f(x)$ is continuous at $x=0$
$\therefore \quad f(0)=\lim _{x \rightarrow 0} f(x) \Rightarrow a=0$