Student Library

One-sided Limits

4 min read

Objectives

  • Define a one-sided limit.
  • Determine the conditions for the existence of a limit

One-sided Limits

  • When we approach a point in a function, we can approach it from two separate directions: from the left and from the right.
  • For example, let us consider the following limit below:
  • When approaching the point , we can use values that are greater than itself.
  • In a graph, it might look something like this:

  • This is also called a limit from the positive direction or limit from the right.
  • Likewise, we can also approach coming from values less than itself
  • In a graph, it might look something like this:

  • This is also called the limit from the negative direction or a limit from the left.

Definition
One-sided Limits

A one-sided limit is a limit approaching a point from a single direction.

  • A limit from the right or a limit from the positive direction approaches the point with values greater than itself. This is expressed as:
  • A limit from the left or a limit from the negative direction approaches the point with values less than itself. This is expressed as:

Existence of a Limit

  • Oftentimes, a limit of a function at a point may note even exist.
  • For example, consider the limit of the following function.
  • If we try to approach the point from the right, we get that the limit is equal to .

  • However, if we try to approach the point from the left, we get that the limit is equal to .
![[oslim-4.gif600]]
  • We see that when we approach them in either side, the limits from both directions don’t match.
  • Therefore, this limit does not exist.

Definition
Existence of a Limit: One-Sided Limits Do Not Match

Given any function , if the one-sided limits at a point do not match, then the limit of at does not exist.

In other words, if:

then, does not exist.

On the other hand, if both one-sided limits match and it is equal to some real number , then the limit of at is .

In other words, if

then, .

  • One other way that the limit does not exist is when the values seem to get bigger and bigger as it gets closer and closer towards a particular point.
  • For example, consider the limit below:
  • For example, if we approach from the right, starting from , we get:
  • Graphically, it may look like this:

  • Likewise, we can also approach from the negative side starting from .
  • Graphically, it may also look like this:

  • No matter how close we get to , it doesn’t seem to approach a single finite value as it just keeps getting larger and larger.
  • Another unique limit is the one below:
  • Taking a look at its graph, we see that as it approaches , it oscillates really fast from to .

  • We can also see that as it approaches from either side, it doesn’t really approach on a fixed, finite value.

Definition
Existence of a Limit: Limit must be a Fixed, Finite Value

Given a function and a point , if:

  • goes unbounded or approaches or as , or
  • doesn’t approach a single, fixed value,

then does not exist.

Graph View