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One-Sided Limits

In the previous section, we discussed the concept of limits and how they can be used to describe the behavior of a function as the input approaches a certain value. However, we did not consider the direction from which the input approaches the value. We could approach the value from the left or the right, and the limit could be different depending on the direction.

A limit with a specific direction is called a one-sided limit.

A graph is shown below, without the function being defined, to drill down on using graphs to understand limits:

In the graph above, we have a function that has a vertical asymptote at . Looking from the graph, it should be clear that going from different directions will give different limits at .

From the left, the function just approaches , but from the right, it hits a vertical asymptote and goes to infinity. We can write both of these as:

The important notational difference here is the and signs. The sign indicates that we are approaching from the left, and the sign indicates that we are approaching from the right.

One good mnemonic is to think of as with a little bit less (i.e. to the left), and as with a little bit more (i.e. to the right).

These are just two examples of one-sided limits, but they can be applied to any function that has a discontinuity or a vertical asymptote. Consider a case where both one-sided limits are equal:

This is then known as the two-sided limit and is the same as the limit we discussed in the previous section.

Generally, when we talk about limits, we are referring to two-sided limits; a function is said to have a limit at a point only if the two one-sided limits are equal.

One-sided limits are nevertheless important when dealing with functions that have vertical asymptotes or other discontinuities.

Summary and Next Steps​

In this section, we introduced the concept of one-sided limits.

Here are the key points to remember:

  1. A one-sided limit is the limit of a function as the input approaches a certain value from a specific direction.
  2. The notation for one-sided limits includes a sign for approaching from the left and a sign for approaching from the right.
  3. One-sided limits are important when dealing with functions that have vertical asymptotes or other discontinuities.
  4. A function is said to have a limit at a point only if the two one-sided limits are equal; that is, the two-sided limit exists.

We have used graphs to understand limits, but in the next section, we will look at another way to estimate limits; through tables.