Question

If $f\left(x\right)=\{\begin{array}{cc}\frac{\sin x^2}{2x\cdot \tan x},& x\ne 0\\ a,& x=0\end{array}$ and $f\left(x\right)$ is continuous at $x=0,$ then value of $a$ is equal to

[1 mark]

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

Answer 1

The correct option is B $\frac{1}{2}$

$f\left(x\right)=\{\begin{array}{cc}\frac{\sin x^2}{2x\cdot \tan x},& x\ne 0\\ a,& x=0\end{array}$
Since $f\left(x\right)$ is continuous at $x=0,$
$\therefore \underset{x\to 0}{lim}f\left(x\right)=f\left(0\right)=a$
$\Rightarrow a=\underset{x\to 0}{lim}\frac{\sin x^2}{2x\cdot \tan x}$
$\Rightarrow a=\underset{x\to 0}{lim}\frac{1}{2}\cdot \frac{\sin \left(x^2\right)}{x^2}\cdot \left(\frac{x}{\tan x}\right)$
$\Rightarrow a=\frac{1}{2}\cdot 1\cdot 1$
$\therefore a=\frac{1}{2}$