Math Help - Limit Discontinuity Question

1. Limit Discontinuity Question

let $f(x) = x^3 + 1$ for $-1 \le x < 0$,
$\frac{1}{2}$ for $x=0$ and
$x^2$ for $0 < x \le 1$.

Prove $\lim_{x \to 0^+} f(x) = 0, \lim_{x \to 0^-} f(x) = 1$ and $\lim_{x \to 0} f(x) = \frac{1}{2}$.

The first two are easy but I get this problem with the third one:

$\forall \epsilon > 0, \exists \delta > 0$ such that $\forall x \in (-\delta, \delta), |f(x) - \frac{1}{2}| < \epsilon$

How do i consider $|f(x) - \frac{1}{2}| < \epsilon$ if f(x) changes on either side of x = 0 ?? Thanks for any help.

2. If $\lim_{x\searrow 0}f(x)=0, \ \lim_{x\nearrow 0}=1$ then $\lim_{x\to 0}f(x)$ does not exist.