Answer
See the below explanation.
Work Step by Step
Since a limit describes how a function behaves near a certain value, the value of a limit is only dependent on how the function is defined $\textbf{locally}$ near a point. So for $f$ and $g$, since they are equal except on a finite set of "bad" points, one may find a small interval containing $a$ that doesn't contain $\textbf{any}$ of these "bad" points(start with some interval and shrink until you remove the closest bad point). Then $f$ and $g$ are $\textbf{identical}$ functions on this interval, and will either have the same limit at $a$ if it exists, or no limit at all.
