I think it's not true.
If then and
Proof by contradiction. Suppose that the limit exists, and that .
We know that . So for each there exists such that whenever .
Choose . Then .
But that means that whenever x is sufficiently close to c, f(x) differs from L by at least 1. That contradicts the assumption that as . (I'll leave it to you to make that last sentence rigorous in terms of epsilons and deltas.)