One-sided limits, and the general limit

Let's look at the graph of log(x) again:
center
Consider the following one-sided limits:They both exist and are both equal to 0.
Indeed:

The limit of a function at a point exists if both the limits from the left and right exist and are equal. This limit is equal to the left and right limits, which are also equal.

You are now ready to learn about Continuity