Lesson Explainer: One-Sided Limits Mathematics

In this explainer, we will learn how to evaluate one-sided limits graphically and algebraically.

We know that the limit of a function describes the behavior of the function near a point. Sometimes, the behavior of a function does not lead to a specific value as the input approaches a limit point.

For instance, consider the function ๐‘“(๐‘ฅ)=|๐‘ฅ|๐‘ฅ which is defined for all real numbers except at ๐‘ฅ=0. If ๐‘ฅ>0, the absolute value on the numerator does not serve any purpose, leading to |๐‘ฅ|=๐‘ฅ. This means that the value of this function is equal to 1 for ๐‘ฅ>0. If ๐‘ฅ<0, then the absolute value takes away the negative sign from the number ๐‘ฅ, while the number in the denominator still has a negative sign. This means that |๐‘ฅ|๐‘ฅ=โˆ’1 if ๐‘ฅ<0. We can describe the behavior of this function near ๐‘ฅ=0 via the following table.

๐‘ฅโˆ’1โˆ’0.5โˆ’0.1โˆ’0.010.010.10.51
๐‘“(๐‘ฅ)โˆ’1โˆ’1โˆ’1โˆ’11111

Based on this function, the value of ๐‘“(๐‘ฅ) behaves differently depending on whether the ๐‘ฅ-value is on the left side or the right side of the limit point ๐‘ฅ=0. In other words, we cannot say that ๐‘“(๐‘ฅ) approaches a specific value as ๐‘ฅ approaches 0.

On the other hand, we can note that there is a pattern in the function value near ๐‘ฅ=0 if we restrict the ๐‘ฅ-values to be on one side of the limit point. For instance, we can just look at the function values near ๐‘ฅ=0 with an additional restriction that ๐‘ฅ<0. In this case, we have only the left half of the table above.

๐‘ฅโˆ’1โˆ’0.5โˆ’0.1โˆ’0.01
๐‘“(๐‘ฅ)โˆ’1โˆ’1โˆ’1โˆ’1

If we only consider this table of values for ๐‘“(๐‘ฅ), we can say that the function value approaches โˆ’1 as ๐‘ฅ approaches 0 from the negative side. We call this the left limit of ๐‘“(๐‘ฅ) at ๐‘ฅ=0. Similarly, we can write the table of function values near ๐‘ฅ=0 for ๐‘ฅ>0.

๐‘ฅ10.50.10.01
๐‘“(๐‘ฅ)1111

This table of function values tells us that ๐‘“(๐‘ฅ) approaches 1 as ๐‘ฅ approaches 0 from the positive side, which is the right limit of ๐‘“(๐‘ฅ) at ๐‘ฅ=0.

Definition: One-Sided Limits

  • If the values of ๐‘“(๐‘ฅ) approach some value ๐ฟ as ๐‘ฅ approaches ๐‘Ž from the negative side (i.e., ๐‘ฅ<๐‘Ž) but not necessarily at ๐‘ฅ=๐‘Ž, then we say that the limit of ๐‘“(๐‘ฅ) as ๐‘ฅ approaches ๐‘Ž from the left is equal to ๐ฟ and we denote this as lim๏—โ†’๏Œบ๏Žช๐‘“(๐‘ฅ)=๐ฟ. This limit is called the left (or left-hand or left-handed) limit of ๐‘“(๐‘ฅ) at ๐‘ฅ=๐‘Ž.
  • Similarly, if the values of ๐‘“(๐‘ฅ) approach some value ๐ฟ as ๐‘ฅ approaches ๐‘Ž from the positive side (i.e., ๐‘ฅ>๐‘Ž) but not necessarily at ๐‘ฅ=๐‘Ž, then we say that the limit of ๐‘“(๐‘ฅ) as ๐‘ฅ approaches ๐‘Ž from the right is equal to ๐ฟ and we denote this as lim๏—โ†’๏Œบ๏Žฉ๐‘“(๐‘ฅ)=๐ฟ. This limit is called the right (or right-hand or right-handed) limit of ๐‘“(๐‘ฅ) at ๐‘ฅ=๐‘Ž.

Using these notations, we can write that limlim๏—โ†’๏Šฆ๏—โ†’๏Šฆ๏Žช๏Žฉ|๐‘ฅ|๐‘ฅ=โˆ’1,|๐‘ฅ|๐‘ฅ=1.

We can also see these one-sided limits when we observe the graph of the function ๐‘“(๐‘ฅ)=|๐‘ฅ|๐‘ฅ.

If we follow the graph of the function on the left side of ๐‘ฅ=0, we approach the point (0,โˆ’1) whose ๐‘ฆ-coordinate is โˆ’1. This tells us that the left limit of this function at ๐‘ฅ=0 is equal to โˆ’1. If we follow the graph on the right side of ๐‘ฅ=0, we approach a point with ๐‘ฆ-coordinate 1, which is the right limit of this function at ๐‘ฅ=0.

In our first example, we will find the one-sided limit of a function from its graph.

Example 1: Finding the One-Sided Limit of a Function from Its Graph at a Point If this Limit Exists

Use the graph shown to find lim๏—โ†’๏Šง๏Žฉ๐‘“(๐‘ฅ).

Answer

We note that 1 under the limit notation has a positive sign in the superscript, which indicates that this is the right limit of this function at ๐‘ฅ=1. We recall that the right limit of a function at ๐‘ฅ=๐‘Ž is the value that ๐‘“(๐‘ฅ) approaches as ๐‘ฅ approaches ๐‘Ž from the right (i.e., ๐‘ฅ>๐‘Ž), but not necessarily at ๐‘ฅ=๐‘Ž. In this example, the limit point is at ๐‘ฅ=1; hence, ๐‘Ž=1.

Recall that a solid point on a graph indicates that the function is defined at that point, while a hollow point on a line indicates that a function does not include that point on the line. As the point (1,6) is solid in the given graph, this indicates that ๐‘“(1)=6, but this is not important for us since the one-sided limit of a function does not consider the value at the limit point. Instead, we need to consider what value ๐‘“(๐‘ฅ) approaches as ๐‘ฅ approaches 1, while ๐‘ฅ>1. Since we are only considering the values of ๐‘ฅ on the right side of 1, we only need to consider the highlighted portion of the graph below.

As we travel toward ๐‘ฅ=1 on the highlighted portion of the graph, we approach a point whose ๐‘ฆ-coordinate is equal to 3. This is the right limit of this function at ๐‘ฅ=1. Hence, we can write lim๏—โ†’๏Šง๏Žฉ๐‘“(๐‘ฅ)=3.

In the next example, we will find the left limit of a function from its graph.

Example 2: Finding the One-Sided Limit of a Function from Its Graph at a Point If this Limit Exists

Determine lim๏—โ†’๏Šฑ๏Šช๏Žช๐‘“(๐‘ฅ).

Answer

We note that โˆ’4 under the limit notation has a negative sign in the superscript, which indicates that this is the left limit of this function at ๐‘ฅ=โˆ’4. We recall that the left limit of a function at ๐‘ฅ=๐‘Ž is the value that ๐‘“(๐‘ฅ) approaches as ๐‘ฅ approaches ๐‘Ž from the left (i.e., ๐‘ฅ<๐‘Ž), but not necessarily at ๐‘ฅ=๐‘Ž. In this example, the limit point is at ๐‘ฅ=โˆ’4; hence, ๐‘Ž=โˆ’4.

We need to find what value ๐‘“(๐‘ฅ) approaches as ๐‘ฅ approaches โˆ’4, while assuming ๐‘ฅ<โˆ’4. The function is not defined at ๐‘ฅ=โˆ’4 since both ends of the line approaching that point are hollow points. However, since we are only considering the values of ๐‘ฅ on the left side of โˆ’4, we only need to consider the highlighted portion of the graph below.

As we travel toward ๐‘ฅ=โˆ’4 on the highlighted portion of the graph, we approach a point whose ๐‘ฆ-coordinate is equal to โˆ’3. This is the left limit of this function at ๐‘ฅ=โˆ’4. Hence, we can write lim๏—โ†’๏Šฑ๏Šช๏Žช๐‘“(๐‘ฅ)=โˆ’3.

Now that we know different types of limits, we should be careful to understand what type of limit the problem is referring to. To distinguish it from the one-sided limits, we often refer to the limit of a function as the regular (or two-sided) limit. In the limit notation, the regular limit does not have the positive or negative sign as a superscript under the limit.

In the first two examples, we found the one-sided limits of functions from the given graphs. In both examples, we were able to find the one-sided limits of the functions although we can clearly see from the graphs that the regular limits of the functions do not exist. This tells us that the one-sided limit can exist even when the regular limit does not exist.

In the next example, we will find the one-sided limit of a function from its graph when the regular limit exists.

Example 3: Finding the One-Sided Limit of a Function from Its Graph at a Point If the Limit Exists

Determine lim๏—โ†’๏Šฑ๏Šฉ๏Žฉ๐‘“(๐‘ฅ).

Answer

We note that โˆ’3 under the limit notation has a positive sign in the superscript, which indicates that this is the right limit of this function at ๐‘ฅ=โˆ’3. We recall that the right limit of a function at ๐‘ฅ=๐‘Ž is the value that ๐‘“(๐‘ฅ) approaches as ๐‘ฅ approaches ๐‘Ž from the right (i.e., ๐‘ฅ>๐‘Ž), but not necessarily at ๐‘ฅ=๐‘Ž. In this example, the limit point is at ๐‘ฅ=โˆ’3; hence, ๐‘Ž=โˆ’3.

Recall that a solid point on a graph indicates that the function is defined at that point, while a hollow point on a line indicates that a function does not include that point on the line. As the point (โˆ’3,โˆ’1) is solid in the given graph, we can see that ๐‘“(โˆ’3)=โˆ’1, but this is not important for us since the one-sided limit of a function does not consider the value at the limit point. Instead, we need to find what value ๐‘“(๐‘ฅ) approaches as ๐‘ฅ approaches โˆ’3, while assuming ๐‘ฅ>โˆ’3. Since we are only considering the values of ๐‘ฅ on the right side of โˆ’3, we only need to consider the highlighted portion of the graph below.

As we travel toward ๐‘ฅ=โˆ’3 on the highlighted portion of the graph, we approach a point whose ๐‘ฆ-coordinate is equal to โˆ’2. This is the right limit of this function at ๐‘ฅ=โˆ’3. Hence, we can write lim๏—โ†’๏Šฑ๏Šฉ๏Žฉ๐‘“(๐‘ฅ)=โˆ’2.

In the previous example, we found the right limit of a function from the given graph. We can see from the graph that the left limit of the function at ๐‘ฅ=โˆ’3 is the same as the right limit. This means that, for the function ๐‘“(๐‘ฅ) whose graph is given in this example, we have limlim๏—โ†’๏Šฑ๏Šฉ๏—โ†’๏Šฑ๏Šฉ๏Žช๏Žฉ๐‘“(๐‘ฅ)=โˆ’2,๐‘“(๐‘ฅ)=โˆ’2.

Furthermore, the regular limit of this function exists at ๐‘ฅ=โˆ’3 and is equal to โˆ’2, which is the same as the value of the one-sided limits. This demonstrates an important connection between one-sided limits and regular limits.

Theorem: Relationship between One-Sided Limits and Regular Limits

Let ๐‘Ž be an interior point of the domain of ๐‘“(๐‘ฅ). Then, the limit of ๐‘“(๐‘ฅ) at ๐‘ฅ=๐‘Ž exists if and only if both the left and right limits of ๐‘“(๐‘ฅ) at ๐‘ฅ=๐‘Ž exist and satisfy limlim๏—โ†’๏Œบ๏—โ†’๏Œบ๏Žช๏Žฉ๐‘“(๐‘ฅ)=๐‘“(๐‘ฅ).

If the limit exists, it is equal to the one-sided limits. That is, limlimlim๏—โ†’๏Œบ๏—โ†’๏Œบ๏—โ†’๏Œบ๐‘“(๐‘ฅ)=๐‘“(๐‘ฅ)=๐‘“(๐‘ฅ).๏Žช๏Žฉ

In particular, this theorem tells us that if the regular limit of a function exists at a point, then all three types of limits (regular, left, and right) take the same value. Since we know how to evaluate the limit of a function, either by direct substitution or by algebraic method, this is a useful method to find the one-sided limit of a function. But we should keep in mind that this method will only work if the (regular) limit of the function exists at the point.

In the next example, we will find the one-sided limit of a piecewise-defined function by finding the regular limit.

Example 4: Finding the One-Sided Limit of a Piecewise-Defined Function Involving Trigonometric Ratios

Find lim๏—โ†’๏Žช๏‘ฝ๏Žฅ๏Žฉ๐‘“(๐‘ฅ) given ๐‘“(๐‘ฅ)=โŽงโŽจโŽฉ7๐‘ฅ+3๐‘ฅ5๐‘ฅ,โˆ’๐œ‹2<๐‘ฅ<0,โˆ’52๐‘ฅ+7,0<๐‘ฅ<๐œ‹2.sinsincos

Answer

We know that the superscript + under the limit indicates that this is the right limit of the function at ๐‘ฅ=โˆ’๐œ‹6. We recall that the one-sided limit of a function takes the same value as the (regular) limit of a function if the (regular) limit exists. If we can find that the limit of ๐‘“(๐‘ฅ) exists at ๐‘ฅ=โˆ’๐œ‹6, this would also be the value of the right limit that we are looking for. So, we will first attempt to find the (regular) limit lim๏—โ†’๏Žช๏‘ฝ๏Žฅ๐‘“(๐‘ฅ).

We know that the limit of a function at ๐‘ฅ=๐‘Ž is the value ๐‘“(๐‘ฅ) approaches when ๐‘ฅ approaches ๐‘Ž. We want to find the limit of ๐‘“(๐‘ฅ) at ๐‘ฅ=โˆ’๐œ‹6, so we want to see what values ๐‘“(๐‘ฅ) take for ๐‘ฅ near โˆ’๐œ‹6. If ๐‘ฅ is sufficiently close to โˆ’๐œ‹6, it will satisfy โˆ’๐œ‹2<๐‘ฅ<0, which is the first condition of the given piecewise function. We can see this by observing the number line.

So, if ๐‘ฅ is near โˆ’๐œ‹6, the function ๐‘“(๐‘ฅ) takes only the first expression of the piecewise function. This means limlimsinsin๏—โ†’๏—โ†’๏Žช๏‘ฝ๏Žฅ๏Žช๏‘ฝ๏Žฅ๐‘“(๐‘ฅ)=7๐‘ฅ+3๐‘ฅ5๐‘ฅ.

This is the limit of a quotient of a function involving the sine function and a polynomial. We know that we can solve this type of a limit by direct substitution as long as the denominator of the quotient is not equal to zero. We can first compute the denominator at ๐‘ฅ=โˆ’๐œ‹6: sinsin5๏€ปโˆ’๐œ‹6๏‡=๏€ผโˆ’5๐œ‹6๏ˆ=โˆ’12.

The denominator is not equal to zero at the limit point, so we can find this limit by direct substitution. This leads to limsinsinsinsin๏—โ†’๏Šฑ๏Ž„๏Šฌ๏Šฑ๏Šซ๏Ž„๏Šฌ๏Šฑ๏Ž„๏Šฌ๏Šฑ๏Šง๏Šจ๏Šฑ๏Šง๏Šจ๏Šฑ๏Šญ๏Ž„๏Šฌ๏Šฉ๏Šจ๏Šฑ๏Šง๏Šจ๏Žช๏‘ฝ๏Žฅ7๐‘ฅ+3๐‘ฅ5๐‘ฅ=7๐‘ฅ+3๏€ป๏‡๏€ป๏‡=7๏€ป๏‡+3๏€ป๏‡=โˆ’=๏€ผโˆ’7๐œ‹6โˆ’32๏ˆร—(โˆ’2)=7๐œ‹3+3.

This gives us the regular limit of ๐‘“(๐‘ฅ) at ๐‘ฅ=โˆ’๐œ‹6. Since the limit exists at this point, we know that the right limit must exist and take the same value as the regular limit. Hence, lim๏—โ†’๏Žช๏‘ฝ๏Žฅ๏Žฉ๐‘“(๐‘ฅ)=3+7๐œ‹3.

In the previous example, we found the one-sided limit of a function by first finding the regular limit of a function. If we examine this method of finding the one-sided limit, we can see that the one-sided limits, just like regular limits, can be eligible for the direct substitution method.

Property: Direct Substitution Methods for One-Sided Limits

We can find the one-sided limit of any of the sum, difference, product, quotient, and composition of any of the functions listed below using direct substitution, as long as the limit point is in the domain of the given function:

  • polynomial or constant function,
  • rational function,
  • power or root function,
  • exponential or logarithmic function,
  • trigonometric function,
  • absolute value function.

If a function is not eligible for direct substitution because the one-sided limit of a function leads to an indeterminate form, we can use the algebraic methods for finding the regular limit to find the one-sided limit.

One-sided limits are often used when discussing limits of a piecewise-defined function at a boundary point. Consider a piecewise-defined function ๐‘“(๐‘ฅ) with subfunctions ๐‘”(๐‘ฅ) and โ„Ž(๐‘ฅ), ๐‘“(๐‘ฅ)=๏ฎ๐‘”(๐‘ฅ)๐‘Ž<๐‘ฅ<๐‘,โ„Ž(๐‘ฅ)๐‘โ‰ค๐‘ฅ<๐‘,ifif for constants ๐‘Ž, ๐‘, ๐‘ satisfying ๐‘Ž<๐‘<๐‘. In this function, ๐‘Ž, ๐‘, and ๐‘ are the boundary points of ๐‘“(๐‘ฅ), so we can focus on the limits of this function at these three points. We can see that the domain of ๐‘“(๐‘ฅ) is ]๐‘Ž,๐‘[, so the limit of ๐‘“(๐‘ฅ) at ๐‘ฅ=๐‘Ž can only be defined on the right side of ๐‘ฅ=๐‘Ž. Hence, in this case, the regular limit of ๐‘“(๐‘ฅ) at ๐‘ฅ=๐‘Ž is the same as the right limit of ๐‘“(๐‘ฅ) at ๐‘ฅ=๐‘Ž. Furthermore, since ๐‘“(๐‘ฅ)=๐‘”(๐‘ฅ) for any values of ๐‘ฅ which is sufficiently close to ๐‘Ž, we can say limlim๏—โ†’๏Œบ๏—โ†’๏Œบ๏Žฉ๏Žฉ๐‘“(๐‘ฅ)=๐‘”(๐‘ฅ).

Similarly, limlim๏—โ†’๏Œผ๏—โ†’๏Œผ๏Žช๏Žช๐‘“(๐‘ฅ)=โ„Ž(๐‘ฅ).

Finally, let us consider the one-sided limits at ๐‘ฅ=๐‘. Since ๐‘“(๐‘ฅ) is defined on either side of ๐‘, both the left and right limits are well defined here. For the left limit, lim๏—โ†’๏Œป๏Žช๐‘“(๐‘ฅ), we are considering values of ๐‘“(๐‘ฅ) for ๐‘ฅ<๐‘. For these values of ๐‘ฅ, we know that ๐‘“(๐‘ฅ)=๐‘”(๐‘ฅ), which leads to limlim๏—โ†’๏Œป๏—โ†’๏Œป๏Žช๏Žช๐‘“(๐‘ฅ)=๐‘”(๐‘ฅ).

Likewise, the right limit of ๐‘“(๐‘ฅ) at ๐‘ฅ=๐‘ is given by limlim๏—โ†’๏Œป๏—โ†’๏Œป๏Žฉ๏Žฉ๐‘“(๐‘ฅ)=โ„Ž(๐‘ฅ).

In particular, if ๐‘”(๐‘ฅ) and โ„Ž(๐‘ฅ) are functions eligible for direct substitution, the left limit of ๐‘“(๐‘ฅ) at ๐‘ฅ=๐‘ would equal ๐‘”(๐‘), and the right limit would equal โ„Ž(๐‘).

In the next example, we will find the one-sided limit of a piecewise function when the limit point is an endpoint of the domain of the function.

Example 5: Discussing the Existence of One-Sided Limits of Piecewise Defined Functions

Discuss the existence of lim๏—โ†’๏Šช๏Žช๐‘“(๐‘ฅ) given ๐‘“(๐‘ฅ)=๏ฒ|๐‘ฅโˆ’1|+2,โˆ’1<๐‘ฅ<2,๐‘ฅ+2๐‘ฅโˆ’8๐‘ฅโˆ’2๐‘ฅ,2<๐‘ฅ<4.๏Šจ๏Šจ

Answer

We know that the superscript โˆ’ under the limit indicates that this is the left limit of the function at ๐‘ฅ=4. We recall that the left limit of a function at ๐‘ฅ=๐‘Ž is the value that ๐‘“(๐‘ฅ) approaches as ๐‘ฅ approaches ๐‘Ž from the left (i.e., ๐‘ฅ<๐‘Ž), but not necessarily at ๐‘ฅ=๐‘Ž. In this example, the limit point is at ๐‘ฅ=4; hence, ๐‘Ž=4.

To find the left limit of the function at ๐‘ฅ=4, we need to find what value ๐‘“(๐‘ฅ) approaches when ๐‘ฅ is near but smaller than 4. If ๐‘ฅ is sufficiently near 4 but smaller than 4, it must satisfy 2<๐‘ฅ<4, which is the second condition of the given piecewise function. For such values of ๐‘ฅ, the function ๐‘“(๐‘ฅ) takes only the second expression of the piecewise function. This means limlim๏—โ†’๏Šช๏—โ†’๏Šช๏Šจ๏Šจ๏Žช๏Žช๐‘“(๐‘ฅ)=๐‘ฅ+2๐‘ฅโˆ’8๐‘ฅโˆ’2๐‘ฅ.

This is a left limit of a rational function. Recall that we can find the one-sided limit of a rational function by direct substitution as long as the denominator does not equal zero at the limit point. We begin by computing the denominator at the limit point ๐‘ฅ=4: 4โˆ’2ร—4=16โˆ’8=8.๏Šจ

The denominator is not equal to zero at the limit point, so we can find this limit by direct substitution. This leads to lim๏—โ†’๏Šช๏Šจ๏Šจ๏Šจ๏Šจ๏Žช๐‘ฅ+2๐‘ฅโˆ’8๐‘ฅโˆ’2๐‘ฅ=4+2ร—4โˆ’84โˆ’2ร—4=16+8โˆ’88=168=2.

Hence, lim๏—โ†’๏Šช๏Žช๐‘“(๐‘ฅ)=2.

We can also find the one-sided limit of a piecewise-defined function at the border between the piecewise intervals. In this case, we need to first choose which of the piecewise expressions to use to find the limit by considering what values of ๐‘ฅ are considered for the one-sided limit.

In the next example, we will consider the one-sided limit of a piecewise function at the border between the piecewise intervals.

Example 6: Finding the One-Sided Limits of a Piecewise-Defined Function

Find lim๏—โ†’๏Šฑ๏Šฏ๏Žช๐‘“(๐‘ฅ) and lim๏—โ†’๏Šฑ๏Šฏ๏Žฉ๐‘“(๐‘ฅ), where ๐‘“(๐‘ฅ)=๏ญ78๐‘ฅ<โˆ’9,โˆ’9๐‘ฅโˆ’7๐‘ฅโ‰ฅโˆ’9.ifif

Answer

We note the superscripts of the positive and negative signs under the limit, which indicate that these are one-sided limits. The limit with the negative sign as the superscript is the left limit, and the other one is the right limit.

Let us begin with the left limit lim๏—โ†’๏Šฑ๏Šฏ๏Žช๐‘“(๐‘ฅ). We recall that the left limit of a function at ๐‘ฅ=๐‘Ž is the value that ๐‘“(๐‘ฅ) approaches as ๐‘ฅ approaches ๐‘Ž from the left (i.e., ๐‘ฅ<๐‘Ž), but not necessarily at ๐‘ฅ=๐‘Ž. In this example, the limit point is at ๐‘ฅ=โˆ’9, so we are assuming ๐‘ฅ<โˆ’9 for the left limit. Note that this is the first condition in the piecewise function. Since the function ๐‘“(๐‘ฅ)=78 for any ๐‘ฅ satisfying this condition, we can say that ๐‘“(๐‘ฅ) approaches 78 as ๐‘ฅ approaches โˆ’9 from the left. Hence, lim๏—โ†’๏Šฑ๏Šฏ๏Žช๐‘“(๐‘ฅ)=78.

Next, let us consider the right limit lim๏—โ†’๏Šฑ๏Šฏ๏Žฉ๐‘“(๐‘ฅ), which assumes that ๐‘ฅ approaches โˆ’9 from the right; that is, ๐‘ฅ>โˆ’9. Since this is the second condition of the given piecewise function, ๐‘“(๐‘ฅ)=โˆ’9๐‘ฅโˆ’7 for any ๐‘ฅ satisfying this condition. This means limlim๏—โ†’๏Šฑ๏Šฏ๏—โ†’๏Šฑ๏Šฏ๏Žฉ๏Žฉ๐‘“(๐‘ฅ)=(โˆ’9๐‘ฅโˆ’7).

This is the right limit of a polynomial function. Recall that we can find one-sided limits of a polynomial function by direct substitution. This leads to lim๏—โ†’๏Šฑ๏Šฏ๏Žฉ(โˆ’9๐‘ฅโˆ’7)=โˆ’9ร—(โˆ’9)โˆ’7=74.

This leads to lim๏—โ†’๏Šฑ๏Šฏ๏Žฉ๐‘“(๐‘ฅ)=74.

Hence, limlim๏—โ†’๏Šฑ๏Šฏ๏—โ†’๏Šฑ๏Šฏ๏Žช๏Žฉ๐‘“(๐‘ฅ)=78,๐‘“(๐‘ฅ)=74.

So far, we have considered examples where one-sided limits exist. It is not true that the one-sided limits of a function must exist. There are two different ways the one-sided limit of a function may not exist. The first case is if the one-sided limit of a function is infinite. For instance, we can consider the function ๐‘“(๐‘ฅ)=1๐‘ฅ at ๐‘ฅ=0, which is graphed below.

We can see from this graph that the right limit of this function is positive infinity, while the left limit of the function is negative infinity. Formally, we can write limlim๏—โ†’๏Šฆ๏—โ†’๏Šฆ๏Žช๏Žฉ1๐‘ฅ=โˆ’โˆž,1๐‘ฅ=โˆž.

But, since infinity is not a number, we can also say that both one-sided limits of this function do not exist.

The second case where the one-sided limit does not exist is due to oscillations in the function. Consider the function whose graph is given below.

From this graph, we can see that the left limit at ๐‘ฅ=โˆ’1 exists and is given by lim๏—โ†’๏Šฑ๏Šง๏Žช=0.25.

As ๐‘ฅ approaches โˆ’1 from the right, we can see that the value ๐‘“(๐‘ฅ) goes back and forth rapidly between the maximum and minimum values. This type of behavior in a function is referred to as โ€œoscillation.โ€ This means that ๐‘“(๐‘ฅ) does not approach a specific value as ๐‘ฅ approaches โˆ’1 from the right, which tells us that the right limit of ๐‘“(๐‘ฅ) at ๐‘ฅ=โˆ’1 does not exist.

In our final example, we will find the one-sided limits of a piecewise-defined function at its boundary where one of the limits does not exist.

Example 7: Finding the One-Sided Limits of a Function

Find lim๏—โ†’๏Šฑ๏Šฏ๏Žช๐‘“(๐‘ฅ) and lim๏—โ†’๏Šฑ๏Šฏ๏Žฉ๐‘“(๐‘ฅ), given that ๐‘“(๐‘ฅ)=๏ฑ๐‘ฅ+9,๐‘ฅ<โˆ’9,1๐‘ฅ+9,๐‘ฅโ‰ฅโˆ’9.ifif

Answer

We note the superscripts of the positive and negative signs under the limit, which indicate that these are one-sided limits. The limit with the negative sign as the superscript is the left limit, and the other one is the right limit.

Let us begin with the left limit lim๏—โ†’๏Šฑ๏Šฏ๏Žช๐‘“(๐‘ฅ). We recall that the left limit of a function at ๐‘ฅ=๐‘Ž is the value that ๐‘“(๐‘ฅ) approaches as ๐‘ฅ approaches ๐‘Ž from the left (i.e., ๐‘ฅ<๐‘Ž), but not necessarily at ๐‘ฅ=๐‘Ž. In this example, the limit point is at ๐‘ฅ=โˆ’9; we are assuming ๐‘ฅ<โˆ’9 for the left limit. Note that this is the first condition in the piecewise function. Since the function ๐‘“(๐‘ฅ)=๐‘ฅ+9 for any ๐‘ฅ satisfying this condition, we can write limlim๏—โ†’๏Šฑ๏Šฏ๏—โ†’๏Šฑ๏Šฏ๏Žช๏Žช๐‘“(๐‘ฅ)=(๐‘ฅ+9).

This is the left limit of a polynomial function. Recall that we can find one-sided limits of a polynomial function by direct substitution. This leads to lim๏—โ†’๏Šฑ๏Šฏ๏Žช(๐‘ฅ+9)=โˆ’9+9=0.

This leads to lim๏—โ†’๏Šฑ๏Šฏ๏Žช๐‘“(๐‘ฅ)=0.

Next, let us consider the right limit lim๏—โ†’๏Šฑ๏Šฏ๏Žฉ๐‘“(๐‘ฅ), which assumes that ๐‘ฅ approaches โˆ’9 from the right; that is, ๐‘ฅ>โˆ’9. Since this is the second condition of the given piecewise function, ๐‘“(๐‘ฅ)=1๐‘ฅ+9 for any ๐‘ฅ satisfying this condition. This means limlim๏—โ†’๏Šฑ๏Šฏ๏—โ†’๏Šฑ๏Šฏ๏Žฉ๏Žฉ๐‘“(๐‘ฅ)=1๐‘ฅ+9.

We note that the denominator ๐‘ฅ+9 equals zero at the limit point ๐‘ฅ=โˆ’9. If ๐‘ฅ is on the right side of 9, then ๐‘ฅ>โˆ’9, which means ๐‘ฅ+9>0. As ๐‘ฅ approaches โˆ’9 from the right, we can see that the denominator becomes smaller yet positive, while the numerator remains as 1. For instance, if ๐‘ฅ=โˆ’8.9, the function value is 1โˆ’8.9+9=10.1=10.

We can construct a table of values to better observe this behavior as ๐‘ฅ approaches โˆ’9 from the right.

๐‘ฅโˆ’8โˆ’8.9โˆ’8.99โˆ’8.999
1๐‘ฅ+91101001โ€Žโ€‰โ€Ž000

We see that the function value increases without bound, which means that the right limit of ๐‘“(๐‘ฅ) at ๐‘ฅ=โˆ’9 is positive infinity. Since infinity is not a number, we can also say that the right limit does not exist at ๐‘ฅ=โˆ’9. Hence, limlimdoesnotexist๏—โ†’๏Šฑ๏Šฏ๏—โ†’๏Šฑ๏Šฏ๏Žช๏Žฉ๐‘“(๐‘ฅ)=0,๐‘“(๐‘ฅ).

Let us finish by recapping a few important concepts from this explainer.

Key Points

  • If the values of ๐‘“(๐‘ฅ) approach some value ๐ฟ as ๐‘ฅ approaches ๐‘Ž from the negative side (i.e., ๐‘ฅ<๐‘Ž) but not necessarily at ๐‘ฅ=๐‘Ž, then we say that the limit of ๐‘“(๐‘ฅ) as ๐‘ฅ approaches ๐‘Ž from the left is equal to ๐ฟ and we denote this as lim๏—โ†’๏Œบ๏Žช๐‘“(๐‘ฅ)=๐ฟ. This limit is called the left (or left-hand or left-handed) limit of ๐‘“(๐‘ฅ) at ๐‘ฅ=๐‘Ž.
  • If the values of ๐‘“(๐‘ฅ) approach some value ๐ฟ as ๐‘ฅ approaches ๐‘Ž from the positive side (i.e., ๐‘ฅ>๐‘Ž) but not necessarily at ๐‘ฅ=๐‘Ž, then we say the limit of ๐‘“(๐‘ฅ) as ๐‘ฅ approaches ๐‘Ž from the right is equal to ๐ฟ and we denote this as lim๏—โ†’๏Œบ๏Žฉ๐‘“(๐‘ฅ)=๐ฟ. This limit is called the right (or right-hand or right-handed) limit of ๐‘“(๐‘ฅ) at ๐‘ฅ=๐‘Ž.
  • The limit of ๐‘“(๐‘ฅ) at ๐‘ฅ=๐‘Ž exists if and only if both the left and right limits of ๐‘“(๐‘ฅ) at ๐‘ฅ=๐‘Ž exist and satisfy limlim๏—โ†’๏Œบ๏—โ†’๏Œบ๏Žช๏Žฉ๐‘“(๐‘ฅ)=๐‘“(๐‘ฅ). If the limit exists, it is equal to the one-sided limits. That is, limlimlim๏—โ†’๏Œบ๏—โ†’๏Œบ๏—โ†’๏Œบ๐‘“(๐‘ฅ)=๐‘“(๐‘ฅ)=๐‘“(๐‘ฅ).๏Žช๏Žฉ

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