Question

If f is an odd function. $\underset{x\to 0}{\lim }f\left(x\right)$exists and is equal to

Answer 1

It is given that,
$\underset{x\to 0}{\lim }f\left(x\right)$ exists.
$\Rightarrow \underset{x\to 0^{-}}{\lim }f\left(x\right)=\underset{x\to 0^{+}}{\lim }f\left(x\right)\Rightarrow \underset{h\to 0}{\lim }f\left(0-h\right)=\underset{h\to 0}{\lim }f\left(0+h\right)\Rightarrow \underset{h\to 0}{\lim }f\left(-h\right)=\underset{h\to 0}{\lim }f\left(h\right)\Rightarrow -\underset{h\to 0}{\lim }f\left(h\right)=\underset{h\to 0}{\lim }f\left(h\right)[\because f(x\left)\mathrm{is}\mathrm{an}\mathrm{odd}\mathrm{function}f\right(-x)=-f(x\left)\right]\Rightarrow 2\underset{h\to 0}{\lim }f\left(h\right)=0\Rightarrow \underset{h\to 0}{\lim }f\left(h\right)=0\underset{x\to 0}{\lim }f\left(x\right)=0$