Question

Does $\underset{x\to 0}{lim}f\left(x\right)$ exist if $f\left(x\right)=\{\begin{array}{c}x,\text{when}x<0\\ 0,\text{when}x=0\\ x^2,\text{when}x>0\end{array}$

Answer 1

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

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

$\underset{x\to 0^{-}}{lim}=\underset{x\to 0}{lim}f(-h)=\underset{h\to 0}{lim}(-h)=0\Rightarrow lim$ exists