Answer
False.
Work Step by Step
False.
Counterexample:
$f(x)=\left\{\begin{array}{ll}
2x+1 & \mathrm{i}\mathrm{f} \ \ x \neq 0\\
1000 & \mathrm{i}\mathrm{f} \ \ x =0.
\end{array}\right.$
Then $\displaystyle \lim_{x\rightarrow 0}f(x)=1$, but $f(0)=1000$.
(As x approaches 0, the function values approach the value 1... but f(0) is not 1.)
