One-sided limits are finite limits where we evaluate the limit from each side individually. The notation we use is
\(\displaystyle{ \lim_{x \to a^-}{f(x)} }\)
evaluate the limit on the left side of \(a\), i.e. values of \(x < a\)
\(\displaystyle{ \lim_{x \to a^+}{f(x)} }\)
evaluate the limit on the right side of \(a\), i.e. values of \(x > a\)
The negative and positive sign that look like exponents on the finite value \(a\) indicate the side that we are looking at.
One of the reasons we need to look at limits on both sides of some number is when we are determining continuity. As you know from the continuity page, one of the requirements for continuity is that the limit at a point must exist. In order for a limit to exist, the limit from the left must be equal to the limit from the right, i.e.
\(\displaystyle{
\lim_{x \to a^-}{f(x)} = \lim_{x \to a^+}{f(x)}
}\)
Notice that we are NOT saying that the function value must be equal to the limit or even that the function need be defined at \(x=a\), only that the limit be equal on both sides of \(a\).
Here is a great video to build your intuition of one-sided limits. He uses an absolute value function
to discuss the idea of one-sided limits and limits that do not exist.
