$ \lim_{x \to x_0^-} f(x) $ exists and is finite $\iff $ $ \forall \epsilon > 0, \exists \delta > 0 $ such that $ x_0 - \delta < x_1, x_2 < x_0 \implies |f(x_1) - f(x_2) | < \epsilon $
I believe I have proven the forward direction using an $ \frac{\epsilon}{2} $ argument. For the backwards direction, I think I need to show that the left limit superior and left limit inferior are equal, but I'm not sure how to go about showing that $f$ is bounded first.