How to Determine if a Limit Does Not Exist Video

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  • 0:03 Limits
  • 0:25 Left-Hand Limits
  • 1:34 Right-Hand Limits
  • 2:39 Two-Sided LImits
  • 3:45 Limits & Graphs
  • 4:59 Lesson Summary
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Lesson Transcript
Instructor: Jenna McDanold

Jenna has two master's degrees in mathematics and has been teaching as an adjunct professor in Chicago for four years.

In this lesson, we'll discuss when a limit does not exist. We'll begin with a description of each type of limit and when that particular type does not exist. Then, we'll use a graph to show how to recognize when a limit does not exist based on the graph of a function ''f.''

Limits

There are three different types of limits: left-hand limits, right-hand limits, and two-sided limits. To determine if a specific limit exists or does not exist, you must first recognize what type of limit you are seeking.

For example, given a function f(x). We will choose a value c from the real number line.

Left-Hand Limits

The left-hand limit is the value that the function f(x) is approaching as x approaches the value of c from the left.

Left Hand Limit

Notice that there is a minus sign as a superscript to the value c in the limit notation. This is what designates the left-hand limit. When determining the value of this limit, we study x values of f that are less than c, and move up towards c.

This limit will only exist when the function is defined for values that are less than c. That is, the left-hand limit will not exist at the left endpoint of the domain for the function f.

For instance, consider the function:

square root function

This function has a domain of (0,∞).

Since 0 is the left endpoint for the domain for this function, the function doesn't exist for any values of x less than 0. This means that the left-hand limit for this function does not exist at the point x = 0.

left hand limit of sqrt x DNE

Note that when a function f has a domain that has negative infinity as its only left endpoint, the left-hand limits for this function will exist at all points in its domain.

Right-Hand Limits

The right-hand limit is the value that the function f(x) is approaching as x approaches the value of c from the right:

right hand limit

Notice that there is a plus sign as a superscript to the value c in the limit notation. This is what designates the right-hand limit. When determining the value of this limit, we study x values of f that are greater than c and move down towards c.

This limit will only exist when the function is defined for values that are greater than c. That is, the right-hand limit will not exist at the right endpoint of the domain for the function f.

For instance, consider the function:

sqrt neg x function

This function has a domain of (-∞,0~).

Since 0 is the right endpoint for the domain for this function, the function doesn't exist for any values of x greater than 0. This means that the right-hand limit for this function does not exist at the point x = 0.

right hand limit of sqrt neg x DNE

Note that when a function f has a domain that has infinity as its only right endpoint, the right-hand limits for this function will exist at all points in its domain.

Two-Sided Limits

The two-sided limit is the typical limit that you see used in math most often and is referred to as just the limit of the function at c. This limit relies on the corresponding one-sided limits to define it.

two sided limit

The two-sided limit exists only if:

  • The left-hand limit exists
  • The right-hand limit exists
  • Both the left-hand limit and the right-hand limit have the same value

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