Section 10.2 Question 2

How do you evaluate a limit from a graph?

In the question before this one, we used a table to observe the output values from a function as the input values approach some value from the left of right. With a little practice, we can evaluate limits using a graph to find the values of a function. Suppose we have the graph of a function like the one below.

10_2_2_01

 

We can use this graph to evaluate the two-sided limit 10_2_2_02.

As with the limits we calculated from tables, we must evaluate the one-sided limits near x = 2. To calculate the limit

10_2_2_03

we must examine the graph at x values that are slightly smaller than x = 2.

10_2_2_04

 

 

Figure 1 – As the x values get closer and closer to 2 from values slightly smaller than 2, the y values approach 4.

In Figure 1, a red dashed vertical line is positioned slightly to the left of 2. The height of the line indicates the y value at that x value. A red dashed horizontal line locates the y value on the graph. As the vertical line moves closer and closer to 2, the horizontal line gets closer and closer to the y value 4. This means the limit as x approaches 2 from the left is 4 or 10_2_2_05.

The same strategy allows us to solve the one-sided limit

10_2_2_06

10_2_2_07

 

Figure 2 – As the x values get closer and closer to 2 from values slightly larger than 2, the y values approach 4.

The red dashed vertical line in Figure 2 locates an x value slightly larger than 2. The red dashed horizontal line gives the corresponding value on the y axis. As the vertical line moves closer and closer to 2, the horizontal line moves closer and closer to 4. In other words, for x values closer and closer to 2, the y values are closer and closer to 4. The limit from the right is 10_2_2_08.

Since the limits from the left and right are both equal to 4, the two-sided limit is also equal to 4,

10_2_2_09

 

Example 4      Find the Limit Graphically

Suppose f (x) is given by the graph below.

10_2_2_10

Evaluate each of the limits below.

a.  10_2_2_11

Solution To evaluate this limit, we need to examine y values on the graph as x gets closer and closer to 1 from the left side of 1. This region of the graph is shown in the graph to the below.

10_2_2_12

Let us locate an x value and its corresponding y value in this region.

10_2_2_13

Notice that as x moves horizontally closer and closer to 1, the corresponding y value moves vertically closer and closer to 1. This tells us that 10_2_2_14 . Notice that the y value at x = 1,  f (1) = 2, is not the same as the limit.

b.  10_2_2_15

Solution In this one sided limit, the x values are on the right side of 1.

10_2_2_16

As the point moves to the left towards , the point moves up vertically towards 1. This means that the closer the point gets to x = 1, the closer the y value gets to 1 or 10_2_2_17.

c.  10_2_2_18

Solution For the two sided limit to exist, the one sided limits must be equal. In this case they are both equal to 1. Since they are both equal to 1, the two sided limit is also equal to 1,

10_2_2_19

Notice that none of these limits have anything to do with the fact that f (1) = 2. This is because we are using x values approaching 1, not equal to 1.


Example 5      Find the Limit Graphically

Suppose f (x) is given by the graph below.

10_2_2_20

Evaluate each of the limits below.

a.  10_2_2_11

Solution To left of x = 1, the graph looks like the graph in Example 1.

10_2_2_21

Notice that as x moves horizontally closer and closer to 1, the corresponding y value moves vertically closer and closer to 1. This tells us that 10_2_2_22.

b.  10_2_2_15

Solution As the point moves to the left towards x = 1, the point moves up vertically towards 2.

 

10_2_2_23

This means that the closer the point gets to , the closer the y value gets to 2 or 10_2_2_24.

c.  10_2_2_18

Solution For the two sided limit to exist, the one sided limits must be equal. In this case, they are not equal. From the left side the limit is equal to 1 and from the right side the limit is equal to 2, so

10_2_2_25

The vertical gap in the graph at is what leads to different values in the one sided limits. In Example 4, there was a horizontal gap at x = 1, but not a vertical gap since the two pieces of the graph come together at x = 1.


 

In each of these examples, we evaluate the one-sided limits to find the two-sided limit. If the one-sided limits are equal to some value, the two-sided limit is equal to the same value. If the one-sided limits do not match, the two-sided limit does not exist. In the next example, we examine a function for which the one-sided limit does not exist.

Example 6      Find the Limit Graphically

Suppose f (x) is given by the graph below.

10_2_2_26

Evaluate the limit 10_2_2_27.

Solution This function has a vertical asymptote at x = 5. The vertical asymptote is shown on the graph as a blue dashed line.

The one-sided limit is a left hand limit. Locate points on the left side of with red dashed lines.

10_2_2_28

As the vertical line gets closer and closer to 5, the horizontal line gets higher and higher. This indicates that the y values do not get closer to any value as x gets closer to 5 from the left. The one-sided limit does not exist.

 

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