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AP Calculus AB: Exam Prep21 chapters | 138 lessons | 6 flashcard sets

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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.''

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.

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

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:

This function has a domain of (0,âˆž).

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.

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.

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

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:

This function has a domain of (-âˆž,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.

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.

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.

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

If the function has both limits defined at a particular *x* value *c* and those values match, then the limit will exist and will be equal to the value of the one-sided limits. If the values of the one-sided limits do not match, then the two-sided limit will no exist.

For instance, consider the function:

This function has the domain:

However, when we test points on either side of 0, what we find is that this function approaches negative infinity as we approach 0 from the left, and it approaches infinity as we approach 0 from the right. This means that the two-sided limit does not exist.

When you graph a function *f*, you can easily tell when the limit of a particular *x* value exists. Here are the rules:

- If the graph has a gap at the
*x*value*c*, then the two-sided limit at that point will not exist. - If the graph has a vertical asymptote and one side of the asymptote goes toward infinity and the other goes toward negative infinity, then the limit does not exist.
- If the graph has a hole at the
*x*value*c*, then the two-sided limit does exist and will be the*y*-coordinate of the hole.

Consider this graph of the function *f*:

In this image, we first look at the point where *x* = 0. We see that there is a gap at that point, and therefore, we know that the two-sided limit does not exist. Each one-sided limit does exist, but they each have different values.

Now let's look at the point where *x* = 1. There is a hole in the graph here, but the limit does exist. In fact, the limit is the value of the y-coordinate for the hole. In this case, the value would be *y* = 1, and therefore the limit as *x* goes to 1 is 1. In mathematical notation, it looks like this:

To recap, let's restate the most important take-aways from this lesson:

- The
**left-hand limit**will not exist at the left endpoint of the domain of the function*f*

- The
**right-hand limit**will not exist the right endpoint of the domain of the function*f*

- The
**two-sided limit**will not exist when either of the one-sided limits do not exist or when their values are not equal.

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AP Calculus AB: Exam Prep21 chapters | 138 lessons | 6 flashcard sets

- Understanding Limits: Using Notation 3:43
- Using a Graph to Define Limits 5:24
- One-Sided Limits and Continuity 4:33
- Infinite Limit: Definition & Rules
- How to Determine the Limits of Functions 5:15
- Understanding the Properties of Limits 4:29
- Squeeze Theorem: Definition and Examples 5:49
- Graphs and Limits: Defining Asymptotes and Infinity 3:29
- How to Determine if a Limit Does Not Exist 5:26
- Go to Limits of Functions

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