Use the graph of given below to determine
The value of the limit isIn this section we use the graph of a function to find limits.
Finding Limits Graphically
Here are some detailed, lecture style videos on graphical limits:
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In this section, functions will be presented graphically. Recall that the graph of a function must pass the vertical line test which states that a vertical line can intersect the graph of a function in at most one point. To understand graphical representations of functions, consider the following graph of a function, .
The graph of a function is created by letting the -coordinate represent the input of
the function and the -coordinate represent the corresponding output, i.e., . The
general form of a point on the graph of is for any input value, , in the domain of
.
Notice that the point is on the graph of , shown above. This means that . Similarly, since the points and are also on the graph, we have and .
We now consider several examples of limits of functions presented graphically.
example 1
For the function, , whose graph is shown below, determine the value of the one-sided limit, Note that this is called a one-sided limit because is approaching from the left hand side.
Solution: Since we know that the -value is approaching and we also know that . Looking at the graph, we can see that for -values slightly less than (to the left of 3), the -values on the graph are very close to . And as the -value moves toward 3, the corresponding -value on the graph moves toward 2. We can express this using limits as
In this example, it is important to note that the open circle at the point indicates that that point is not on the graph. Furthermore, there are no points on the graph with -coordinate equal to 3, which means that is undefined. Thus, we see in this example that a function can have a limit as approaches a value that is not in the domain of the function. Interactivity note: if you click on the graph, a dot will appear on the graph. You can then drag that dot along the curve with your mouse.
example 2 For the function, , whose graph is shown below, determine the
value of the two-sided limit, Note that this is called a two-sided limit
because can approach from either the left hand side or the right hand
side.
Solution: Since we know that the -value is approaching and we also know that can be on either side of (but not equal to 3). Looking at the graph, we can see that for -values either slightly less than or slightly greater than , the -values on the graph are very close to . Thus,
In this example it is important to observe that, even though the function value at is ,
as indicated by the dot at the point , the limit of the function as approaches is and
not . The limit of a function is determined by the behavior of the function near the
indicated -value and not at that -value.
example 3 For the function, , whose graph is shown below, find the one-sided limits,
and discuss the two-sided limit,
Solution: As in example 1, we can determine the value of the left hand limit, , by observing that if is on the left side of 1 and very close to 1 then the -values on the graph are very close to -1 and hence,
For the right hand limit , we inspect the -values on the graph that correspond to -values on the right side of 1 and very close to . We see that these -values are very close to 2, hence
We see that the one-sided limits as approaches 1 have different values, and we therefore conclude that the two-sided limit, , does not exist. We write
In general, when the one-sided limits are different then the corresponding two-sided limit does not exist.
example 4 For the function, , whose graph is shown below, find the one-sided limit,
Solution: From the graph, we can see that as the -values approach 3 from the right hand side, the -values decrease without bound. In this case we write
To describe the phenomenon of an infinite limit as approaches a finite value (in this case, 3), we say that the line is a vertical asymptote for the graph of .
example 5 For the function, , whose graph is shown below, find the limit, This is an
example of a limit at infinity and it is used to help us describe the end behavior of
the function.
Solution: Since is approaching infinity, we look for a pattern on the right end of the graph. From the graph, we can see that the -values are getting closer and closer to as the values increase without bound. We can conclude that
and we can describe the end behavior by saying that the line is a horizontal asymptote for the graph of .
Here is a problem that encompasses all of the ideas in this section.
(problem 6)
Use the above graph of to answer the following questions.
What are the following function values (undefined is a possibility)? Find the following one-sided limits. (You can click on the graph and drag a point along it.) Type infinity for and -infinity for . Find the following limits at infinity. (Click on the grid and drag it if necessary. Press the home button on the right to reset the grid.)
In the next few problems, we will reverse the situation. Information about the limits and values of a function are given and you are asked to construct the graph of the function. Many answers are possible, but if your answer is correct, then your graph will have all of the properties listed.