Question

A function f(x) is defined as
$\{\begin{array}{cc}\pi \frac{sinx}{x}& x\ne 0\\ \frac{22}{7}& x=0\end{array}$

then f(x) at x = 0 is

• A. Neither continuous nor differentiable
• B. Continuous but not differentiable
• C. Continuous and differentiable
• D. Not defined at x = 0

Answer 1

The correct option is A Neither continuous nor differentiable

LHL $li{m}_{x\to 0^{-}}f\left(x\right)=li{m}_{x\to 0^{-}}\frac{\pi sinx}{x}\to \left[\frac{0}{0}form\right]$

Applying L Hospital rule,

$li{m}_{x\to 0^{-}}f\left(x\right)={lim}_{x\to 0^{-}}\pi cosx=\pi$ ...(i)

Similarly,

RHL $li{m}_{x\to 0^{+}}f\left(x\right)=\pi$ ...(2)

Also,

RHL $f\left(0\right)=\frac{22}{7}$ ...(3)

So, from (1), (2), & (3),

LHL = RHL $\ne$ functional value

$\Rightarrow f\left(x\right)$ is not continuous at x =0
Note : $\pi \ne \frac{22}{7}$. It is approximated to $\frac{22}{7}$