Answer
Yes, see explanations.
Work Step by Step
If we know $\lim_{x\to a^+}f(x)=L_1$ and we know $\lim_{x\to a^-}f(x)=L_2$, there are two cases: (i) if $L_1=L_2=L$, then we know $\lim_{x\to a}f(x)=L$; (ii) if $L_1\ne L_2$, the limit $\lim_{x\to a}f(x)$ does not exist.
Thus, the answer is Yes.
