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 which is defined for all real numbers except at . If , the absolute value on the numerator does not serve any purpose, leading to . This means that the value of this function is equal to 1 for . If , then the absolute value takes away the negative sign from the number , while the number in the denominator still has a negative sign. This means that if . We can describe the behavior of this function near via the following table.
| 0.01 | 0.1 | 0.5 | 1 | |||||
| 1 | 1 | 1 | 1 |
Based on this function, the value of behaves differently depending on whether the -value is on the left side or the right side of the limit point . In other words, we cannot say that approaches a specific value as approaches 0.
On the other hand, we can note that there is a pattern in the function value near if we restrict the -values to be on one side of the limit point. For instance, we can just look at the function values near with an additional restriction that . In this case, we have only the left half of the table above.
If we only consider this table of values for , we can say that the function value approaches as approaches 0 from the negative side. We call this the left limit of at . Similarly, we can write the table of function values near for .
| 1 | 0.5 | 0.1 | 0.01 | |
| 1 | 1 | 1 | 1 |
This table of function values tells us that approaches 1 as approaches 0 from the positive side, which is the right limit of at .
Definition: One-Sided Limits
- If the values of approach some value as approaches from the negative side (i.e., ) but not necessarily at , then we say that the limit of as approaches from the left is equal to and we denote this as This limit is called the left (or left-hand or left-handed) limit of at .
- Similarly, if the values of approach some value as approaches from the positive side (i.e., ) but not necessarily at , then we say that the limit of as approaches from the right is equal to and we denote this as This limit is called the right (or right-hand or right-handed) limit of at .
Using these notations, we can write that
We can also see these one-sided limits when we observe the graph of the function .
If we follow the graph of the function on the left side of , we approach the point whose -coordinate is . This tells us that the left limit of this function at is equal to . If we follow the graph on the right side of , we approach a point with -coordinate 1, which is the right limit of this function at .
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 .
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 . We recall that the right limit of a function at is the value that approaches as approaches from the right (i.e., ), but not necessarily at . In this example, the limit point is at ; hence, .
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 is solid in the given graph, this indicates that , 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 approaches as approaches 1, while . Since we are only considering the values of on the right side of 1, we only need to consider the highlighted portion of the graph below.
As we travel toward on the highlighted portion of the graph, we approach a point whose -coordinate is equal to 3. This is the right limit of this function at . Hence, we can write
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 .
Answer
We note that under the limit notation has a negative sign in the superscript, which indicates that this is the left limit of this function at . We recall that the left limit of a function at is the value that approaches as approaches from the left (i.e., ), but not necessarily at . In this example, the limit point is at ; hence, .
We need to find what value approaches as approaches , while assuming . The function is not defined at since both ends of the line approaching that point are hollow points. However, since we are only considering the values of on the left side of , we only need to consider the highlighted portion of the graph below.
As we travel toward on the highlighted portion of the graph, we approach a point whose -coordinate is equal to . This is the left limit of this function at . Hence, we can write
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 .
Answer
We note that under the limit notation has a positive sign in the superscript, which indicates that this is the right limit of this function at . We recall that the right limit of a function at is the value that approaches as approaches from the right (i.e., ), but not necessarily at . In this example, the limit point is at ; hence, .
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 is solid in the given graph, we can see that , 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 approaches as approaches , while assuming . Since we are only considering the values of on the right side of , we only need to consider the highlighted portion of the graph below.
As we travel toward on the highlighted portion of the graph, we approach a point whose -coordinate is equal to . This is the right limit of this function at . Hence, we can write
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 is the same as the right limit. This means that, for the function whose graph is given in this example, we have
Furthermore, the regular limit of this function exists at and is equal to , 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 be an interior point of the domain of . Then, the limit of at exists if and only if both the left and right limits of at exist and satisfy
If the limit exists, it is equal to the one-sided limits. That is,
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 given
Answer
We know that the superscript under the limit indicates that this is the right limit of the function at . 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 exists at , 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
We know that the limit of a function at is the value approaches when approaches . We want to find the limit of at , so we want to see what values take for near . If is sufficiently close to , it will satisfy , which is the first condition of the given piecewise function. We can see this by observing the number line.
So, if is near , the function takes only the first expression of the piecewise function. This means
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 :
The denominator is not equal to zero at the limit point, so we can find this limit by direct substitution. This leads to
This gives us the regular limit of at . 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,
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 with subfunctions and , for constants , , satisfying . In this function, , , and are the boundary points of , so we can focus on the limits of this function at these three points. We can see that the domain of is , so the limit of at can only be defined on the right side of . Hence, in this case, the regular limit of at is the same as the right limit of at . Furthermore, since for any values of which is sufficiently close to , we can say
Similarly,
Finally, let us consider the one-sided limits at . Since is defined on either side of , both the left and right limits are well defined here. For the left limit, , we are considering values of for . For these values of , we know that , which leads to
Likewise, the right limit of at is given by
In particular, if and are functions eligible for direct substitution, the left limit of at would equal , and the right limit would equal .
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 given
Answer
We know that the superscript under the limit indicates that this is the left limit of the function at . We recall that the left limit of a function at is the value that approaches as approaches from the left (i.e., ), but not necessarily at . In this example, the limit point is at ; hence, .
To find the left limit of the function at , we need to find what value approaches when is near but smaller than 4. If is sufficiently near 4 but smaller than 4, it must satisfy , which is the second condition of the given piecewise function. For such values of , the function takes only the second expression of the piecewise function. This means
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 :
The denominator is not equal to zero at the limit point, so we can find this limit by direct substitution. This leads to
Hence,
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 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 and , where
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 . We recall that the left limit of a function at is the value that approaches as approaches from the left (i.e., ), but not necessarily at . In this example, the limit point is at , so we are assuming for the left limit. Note that this is the first condition in the piecewise function. Since the function for any satisfying this condition, we can say that approaches 78 as approaches from the left. Hence,
Next, let us consider the right limit , which assumes that approaches from the right; that is, . Since this is the second condition of the given piecewise function, for any satisfying this condition. This means
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
This leads to
Hence,
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 at , 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
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 exists and is given by
As approaches from the right, we can see that the value goes back and forth rapidly between the maximum and minimum values. This type of behavior in a function is referred to as “oscillation.” This means that does not approach a specific value as approaches from the right, which tells us that the right limit of at 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 and , given that
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 . We recall that the left limit of a function at is the value that approaches as approaches from the left (i.e., ), but not necessarily at . In this example, the limit point is at ; we are assuming for the left limit. Note that this is the first condition in the piecewise function. Since the function for any satisfying this condition, we can write
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
This leads to
Next, let us consider the right limit , which assumes that approaches from the right; that is, . Since this is the second condition of the given piecewise function, for any satisfying this condition. This means
We note that the denominator equals zero at the limit point . If is on the right side of 9, then , which means . As approaches from the right, we can see that the denominator becomes smaller yet positive, while the numerator remains as 1. For instance, if , the function value is
We can construct a table of values to better observe this behavior as approaches from the right.
| 1 | 10 | 100 | 1 000 |
We see that the function value increases without bound, which means that the right limit of at is positive infinity. Since infinity is not a number, we can also say that the right limit does not exist at . Hence,
Let us finish by recapping a few important concepts from this explainer.
Key Points
- If the values of approach some value as approaches from the negative side (i.e., ) but not necessarily at , then we say that the limit of as approaches from the left is equal to and we denote this as This limit is called the left (or left-hand or left-handed) limit of at .
- If the values of approach some value as approaches from the positive side (i.e., ) but not necessarily at , then we say the limit of as approaches from the right is equal to and we denote this as This limit is called the right (or right-hand or right-handed) limit of at .
- The limit of at exists if and only if both the left and right limits of at exist and satisfy If the limit exists, it is equal to the one-sided limits. That is,
