Question

If $f\left(x\right)=\{\begin{array}{c}\frac{\tan x}{x},x\ne 0\\ 1,x=0\end{array}$ then $f\left(x\right)$ at $x=0$ is:

• A. differentiable
• B. Continuous
• C. Discontinuous
• D. Not differentiable

Answer 1

Answer: (A), (B)

The correct options are
A differentiable
B Continuous

Here ,$f^{\prime }\left(0\right)=\underset{h\to 0}{lim}\frac{f(0+h)-f\left(0\right)}{h}=\underset{h\to 0}{lim}\frac{\frac{\tan h}{h}-1}{h}$

$=\underset{h\to 0}{lim}\frac{1-1}{h}=0\left(finite\right)$

Hence, $f\left(x\right)$ is differentiable at $x=0$.
$\Rightarrow f\left(x\right)$ is continuous at $x=0$