One-sided Limits

We will begin studying this by a discussion about the function . The domain of this function is . Suppose we have to find the limit of the function as x approaches 0. Before we proceed, we should note that the function is defined only for nonnegative numbers, which are only on the right of 0. That means that we cannot find an open interval containing 0 on which the function is defined, while finding the limit of the function at 0 requires it to be defined on such an interval, but not necessarily at 0, as the definition of a limit stated.

Does that mean that the function does not have a limit at 0? The function actually gets closer to a number, which is 0, when x gets closer to 0. However, x cannot approach 0 except from the right. That’s why we need to define the one-sided limits.

Definition

Let f be a function defined at least on an open interval . We say that the right-hand limit of f at a is equal to L, or in symbols,

,

if corresponding to each positive number  there is some positive number  such that:

We say that the left-hand limit of the f at b is equal to L, or in symbols,

,

if corresponding to each positive number  there is some positive number  such that:

You can notice that  in the first part of the definition implies that . So, x approaches c from the right only, while in the second part, we have , which implies that , which means that x approaches c from the left only.

Example

Prove that:

Solution

In the case of the functions defined on an open interval, which contains the point that we are calculating the limit at, the following theorem is true.

Theorem

If f is a function defined on an open interval I, where . Then:

.

Note that the theorems on limits in the last section applies also to one-sided limits.