Lesson Explainer: Limits from Tables and Graphs Mathematics • Higher Education

In this explainer, we will learn how to evaluate the limit of a function using tables and graphs.

The limit of a function at a point gives us information about the outputs of that function near (but not at) the point. The formal definition of the limit of a function is as follows.

Definition: Limit of a Function

If the values of 𝑓(π‘₯) approach some value of 𝐿 as the values of π‘₯ approach π‘Ž (from both sides) but not necessarily when π‘₯=π‘Ž, then we say the limit of 𝑓(π‘₯) as π‘₯ approaches π‘Ž is equal to 𝐿 and we denote this as limο—β†’οŒΊπ‘“(π‘₯)=𝐿.

To determine the limit of a function at a point, we need to investigate the outputs of the function as the values of π‘₯ get arbitrarily close to the point (from both sides). We can investigate the outputs of the function in two different ways.

First, we can estimate the limit of a function from its graph. We recall, in the graph of a function, given by 𝑦=𝑓(π‘₯), that a point on the curve has the form (π‘₯, 𝑓(π‘₯)), so the 𝑦-coordinates of the points tell us the output of the function. Therefore, we can investigate the limit of a function from its graph by considering the shape of its graph on both sides of the point. For example, consider the following graph of 𝑓(π‘₯)=π‘₯.

We can determine limο—β†’οŠ¦οŠ¨π‘₯ by considering the outputs of the function on both sides of 0.

From the diagram, we can see that 𝑓(2)=4.

To determine the value of the limit, we need the values of π‘₯ to approach 0 and we can do this by considering more input values.

From the graph, we can see that taking input values of π‘₯ closer to 0 from the right makes the outputs approach 0. To investigate the limit, we need to check that this happens on both sides. We can do this graphically.

Once again, as the input values approach 0 from the left, the graph suggests the output values are approaching 0. Therefore, the graph suggests limο—β†’οŠ¦οŠ¨π‘₯=0.

In the above method for investigating a limit, we needed the graph of the function to estimate its outputs. However, we can find the exact outputs of a function by evaluation. Therefore, we can investigate the limit of a function by taking sample points of the function.

For example, we can also investigate limο—β†’οŠ¦οŠ¨π‘₯ by taking sample points. We need to check whether the values of 𝑓(π‘₯) approach some value of 𝐿 as the values of π‘₯ approach 0 (from both sides). We can start with values of π‘₯>0 and we know 𝑓(1)=1=1. We need the input values of π‘₯ to approach 0. We can see that 𝑓(0.1)=0.01 and 𝑓(0.01)=0.0001, which gives us the following table.

π‘₯0←0.010.11
𝑓(π‘₯)←0.00010.011

Remember, when investigating a limit, we need to check both sides. We have 𝑓(βˆ’1)=1, 𝑓(βˆ’0.1)=0.01, and 𝑓(βˆ’0.01)=0.0001. We can add this information to our table.

π‘₯βˆ’1βˆ’0.1βˆ’0.01β†’0←0.010.11
𝑓(π‘₯)10.010.0001→←0.00010.011

As the values of π‘₯ approach 0 from either side, the outputs of the function are approaching 0. Hence, the table suggests limο—β†’οŠ¦οŠ¨π‘₯=0.

Let’s see an example of how to use a table of outputs to estimate a limit.

Example 1: Finding the Limit of a Function from a Table

Estimate limο—β†’οŠ±οŠ¨π‘“(π‘₯) from the given table.

π‘₯βˆ’2.1βˆ’2.01βˆ’2.001β†’βˆ’2β†βˆ’1.999βˆ’1.99βˆ’1.9
𝑓(π‘₯)36.936.0936.009→←35.99135.9135.1

Answer

We write limο—β†’οŠ±οŠ¨π‘“(π‘₯)=𝐿 if, as the input values of π‘₯ get arbitrarily close to βˆ’2, from both sides, the output values approach 𝐿.

In the table, we can see that as the input values of π‘₯ approach βˆ’2 from the right, the output values form the sequence 35.1, 35.91, 35.991, which appears to be approaching 36. One way of seeing this is to consider the difference between each value and 36:

π‘₯βˆ’2β†βˆ’1.999βˆ’1.99βˆ’1.9
𝑓(π‘₯)←35.99135.9135.1
𝑓(π‘₯)βˆ’36βˆ’0.009βˆ’0.09βˆ’0.9

We can do the same with the values of π‘₯ from the left.

π‘₯βˆ’2.1βˆ’2.01βˆ’2.001β†’βˆ’2β†βˆ’1.999βˆ’1.99βˆ’1.9
𝑓(π‘₯)36.936.0936.009→←35.99135.9135.1
𝑓(π‘₯)βˆ’360.90.090.009βˆ’0.009βˆ’0.09βˆ’0.9

Once again, the table suggests these values are approaching 36. Hence, as the outputs approach 36 as the values of π‘₯ approach βˆ’2 from either direction, the table suggests limο—β†’οŠ±οŠ¨π‘“(π‘₯)=36.

In our next example, we will investigate the limit of a function from its graph.

Example 2: Finding the Limit of a Function from Its Graph

If the graph shown represents the function 𝑓(π‘₯)=π‘₯βˆ’3, determine limο—β†’οŠ±οŠ§π‘“(π‘₯).

Answer

We recall that the notation limο—β†’οŠ±οŠ§π‘“(π‘₯)=𝐿 means that as our values of π‘₯ get arbitrarily close to βˆ’1, from either direction, the outputs of the function must approach 𝐿. We can check each direction separately from the graph. First, we can see that 𝑓(1)=βˆ’2.

We keep choosing input values to approach βˆ’1. For example, 𝑓(0)=βˆ’3 and 𝑓(βˆ’0.5)=βˆ’3.5.

As we approach the input value of βˆ’1 in this manner, the outputs of the function approach βˆ’4.

To investigate the limit as π‘₯ approaches βˆ’1, we need to check the outputs on the other side of π‘₯=βˆ’1. We can do this by using the graph.

Once again, as our input values approach βˆ’1 from the left, the outputs of the function approach βˆ’4.

As the inputs of the function approach βˆ’1 from either side, we can see that the outputs approach βˆ’4. Hence, the graph indicates limο—β†’οŠ±οŠ§π‘“(π‘₯)=βˆ’4.

In our next example, we will determine the limit of a nonlinear function from its graph.

Example 3: Finding the Limit of a Function from Its Graph If the Limit Exists

Determine the limit as π‘₯β†’2 of the function represented by the graph.

Answer

We recall that saying the limit of a function as π‘₯β†’2 is equal to some value 𝐿 means that as our values of π‘₯ get arbitrarily close to 2, from either direction, the outputs of our function must approach 𝐿. We can determine the outputs of the function from its graph, so we can determine this limit by considering the outputs from the graph on either side of π‘₯=2.

We can see from the graph that as the values of π‘₯ get closer and closer to 2 from the right, the output values of the function appear to approach 3. We can do the same from the left side.

Once again, as our values of π‘₯ get arbitrarily close to 2 from the left, the output values of the function appear to approach 3. Since the outputs appear to approach 3 as π‘₯ approaches 2 from either side, limο—β†’οŠ¨π‘“(π‘₯)=3.

In our next example, we will determine the limit of a function from its graph when the function is not defined at the limit point.

Example 4: Finding the Limit of a Function from Its Graph at a Point of Removable Discontinuity If the Limit Exists

Determine limο—β†’οŠ¨π‘“(π‘₯) if it exists.

Answer

We write limο—β†’οŠ¨π‘“(π‘₯)=𝐿 if the values of 𝑓(π‘₯) approach some value of 𝐿 as the values of π‘₯ approach 2, from both sides, but not necessarily when π‘₯=2. We can investigate the values of 𝑓(π‘₯) as π‘₯ approaches 2 from either side by using the graph. Starting from the left, we get the following.

As the values of π‘₯ approach 2 from the left, the outputs of the function appear to get closer to 3. We can do the same from the right.

This time, as the values of π‘₯ approach 2 from the right, the outputs of the function approach 3. It is important to reiterate that while 𝑓(2) is undefined, as shown by the hollow dot at (2,3), this does not affect the limit since we are only interested in the outputs arbitrarily close to 2 and not at π‘₯=2.

Therefore, the graph suggests limο—β†’οŠ¨π‘“(π‘₯)=3.

In our final example, we will see that the value of the function at a point can be different from its limit at that point.

Example 5: Finding the Limit of a Function from Its Graph If the Limit Exists

Using the graph shown, determine limο—β†’οŠ©π‘“(π‘₯).

Answer

We write that limο—β†’οŠ©π‘“(π‘₯)=𝐿 if the values of 𝑓(π‘₯) approach some value, 𝐿, as the values of π‘₯ approach 3 (from both sides) but not necessarily when π‘₯=3. We can investigate the values of 𝑓(π‘₯) as π‘₯ approaches 3 from either side by using the graph. On the right side of π‘₯=3, we have the following.

As the input values of π‘₯ approach 3 from the right, the outputs of the function approach 2. We can then do the same from the left.

As the input values of π‘₯ approach 3 from the left, the outputs of the function, once again, approach 2.

Note that while the graph shows that 𝑓(3)=βˆ’5, since this is where the solid dot occurs when π‘₯=3, it is important to remember that the limit of any function at π‘₯=π‘Ž is determined by its outputs arbitrarily close to π‘Ž and not when π‘₯=π‘Ž.

Hence, the graph suggests limο—β†’οŠ©π‘“(π‘₯)=2.

Let’s finish by recapping some of the important points of this explainer.

Key Points

  • If the values of 𝑓(π‘₯) approach some value of 𝐿 as the values of π‘₯ approach π‘Ž (from both sides) but not necessarily when π‘₯=π‘Ž, then we say limο—β†’οŒΊπ‘“(π‘₯)=𝐿.
  • The value of the function at π‘Ž does not affect its limit as π‘₯ approaches π‘Ž.
  • We can investigate the limit of a function at π‘₯=π‘Ž from its graph by considering what happens to the curve on either side of π‘₯=π‘Ž.
  • We can investigate the limit of a function by taking sample points above and below the value of the limit we are taking. This is often represented by a table of the function’s outputs.

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