Question

If $f$ is an odd function and if $\underset{x\to 0}{lim}f\left(x\right)$ exists, then the value of $\underset{x\to 0}{lim}f\left(x\right)$ ?

• A. $0$
• B. $-1$
• C. $1$
• D. Does not exist

Answer 1

Answer: (A)

$0$

It is given that $\underset{x\to 0}{lim}f\left(x\right)$ exists.

$\therefore \underset{x\to 0^{-}}{lim}f\left(x\right)=\underset{x\to 0^{+}}{lim}f\left(x\right)$

$\Rightarrow \underset{h\to 0}{lim}f(0-h)=\underset{h\to 0}{lim}f(0+h)$

$\Rightarrow \underset{h\to 0}{lim}f(-h)=\underset{h\to 0}{lim}f\left(h\right)$

$\Rightarrow -\underset{h\to 0}{lim}f\left(h\right)=\underset{h\to 0}{lim}f\left(h\right)[\because f(x)$ is odd $\therefore f(-h)=-f\left(h\right)]$

$\Rightarrow 2\underset{h\to 0}{lim}f\left(h\right)=0$

$\Rightarrow {lim}_{h\to 0}f\left(h\right)=0$

$\Rightarrow {lim}_{x\to 0}f\left(x\right)=0$