In Section 1.1 we used a function, , to model the location of a moving object at a given time. Functions can model other interesting phenomena, such as the rate at which an automobile consumes gasoline at a given velocity, or the reaction of a patient to a given dosage of a drug. We can use calculus to study how a function value changes in response to changes in the input variable.
Note that the average velocity is a function of . That is, the function tells us the average velocity of the ball on the interval from to . To find the instantaneous velocity of the ball when , we need to know what happens to as gets closer and closer to . But also notice that is not defined, because it leads to the quotient .
This is where the notion of a limit comes in. By using a limit, we can investigate the behavior of as gets arbitrarily close, but not equal, to . We first use the graph of a function to explore points where interesting behavior occurs.
As gets closer and closer (but not equal) to , does get as close as we want to a single value? If such a value exists, enter it. If no such value exists, enter DNE.
Limits give us a way to identify a trend in the values of a function as its input variable approaches a particular value of interest. We need a precise understanding of what it means to say “a function has limit as approaches .” To begin, think about a recent example.
In Preview Activity 1.2.1, we saw that as gets closer and closer (but not equal) to 0, gets as close as we want to the value 4. At first, this may feel counterintuitive, because the value of is , not . But limits describe the behavior of a function arbitrarily close to a fixed input, and the value of the function at the fixed input does not matter. More formally, 1
What follows here is not what mathematicians consider the formal definition of a limit. To be completely precise, it is necessary to quantify both what it means to say “as close to as we like” and “sufficiently close to .” That can be accomplished through what is traditionally called the epsilon-delta definition of limits. The definition presented here is sufficient for the purposes of this text.
provided that we can make as close to as we like by taking sufficiently close (but not equal) to . If we cannot make as close to a single value as we would like as approaches , then we say that does not have a limit as approaches .
When working from a graph, it suffices to ask if the function approaches a single value from each side of the fixed input. The function value at the fixed input is irrelevant. This reasoning explains the values of the three limits stated above.
However, does not have a limit as . There is a jump in the graph at . If we approach from the left, the function values tend to get close to 3, but if we approach from the right, the function values get close to 2. There is no single number that all of these function values approach. This is why the limit of does not exist at .
For any function , there are typically three ways to answer the question “does have a limit at , and if so, what is the limit?” The first is to reason graphically as we have just done with the example from Preview Activity 1.2.1. If we have a formula for , there are two additional possibilities:
Evaluate the function at a sequence of inputs that approach on either side (typically using some sort of computing technology), and ask if the sequence of outputs seems to approach a single value.
Use the algebraic form of the function to understand the trend in its output values as the input values approach .
The first approach produces only an approximation of the value of the limit, while the latter can often be used to determine the limit exactly.
For each of the following functions, we’d like to know whether or not the function has a limit at the stated -values. Use both numerical and algebraic approaches to investigate and, if possible, estimate or determine the value of the limit. Compare the results with a careful graph of the function on an interval containing the points of interest.
a. We first construct a graph of along with tables of values near and .
Table1.2.4.Table of values near .
Table1.2.5.Table of values near .
Figure1.2.6.Plot of on .
From Table 1.2.4, it appears that we can make as close as we want to 3 by taking sufficiently close to , which suggests that . This is also consistent with the graph of . To see this a bit more rigorously and from an algebraic point of view, consider the formula for :. As ,, and , so as , the numerator of tends to 3 and the denominator tends to 1, hence .
The situation is more complicated when , because is not defined. If we try to use a similar algebraic argument regarding the numerator and denominator, we observe that as ,, and , so as , the numerator and denominator of both tend to 0. We call an indeterminate form. This tells us that there is somehow more work to do. From Table 1.2.5 and Figure 1.2.6, it appears that should have a limit of at .
To see algebraically why this is the case, observe that
.
It is important to observe that, since we are taking the limit as , we are considering values that are close, but not equal, to . Because we never actually allow to equal , the quotient has value 1 for every possible value of . Thus, we can simplify the most recent expression above, and find that
.
This limit is now easy to determine, and its value clearly is . Thus, from several points of view we’ve seen that .
b. Next we turn to the function , and construct two tables and a graph.
Table1.2.7.Table of values near .
Table1.2.8.Table of values near .
Figure1.2.9.Plot of on .
First, as , it appears from the table values that the function is approaching a number between and . From the graph it appears that as . The exact value of is , which is approximately 0.8660254038. This is convincing evidence that
.
As , we observe that does not behave in an elementary way. When is positive and approaching zero, we are dividing by smaller and smaller positive values, and increases without bound. When is negative and approaching zero, decreases without bound. In this sense, as we get close to , the inputs to the sine function are growing rapidly, and this leads to increasingly rapid oscillations in the graph of betweem and . If we plot the function with a graphing utility and then zoom in on , we see that the function never settles down to a single value near the origin, which suggests that does not have a limit at .
How do we reconcile the graph with the righthand table above, which seems to suggest that the limit of as approaches may in fact be ? The data misleads us because of the special nature of the sequence of input values . When we evaluate , we get for each positive integer value of . But if we take a different sequence of values approaching zero, say , then we find that
.
That sequence of function values suggests that the value of the limit is . Clearly the function cannot have two different values for the limit, so has no limit as .
An important lesson to take from Example 1.2.3 is that tables can be misleading when determining the value of a limit. While a table of values is useful for investigating the possible value of a limit, we should also use other tools to confirm the value.
Estimate the value of each of the following limits by constructing appropriate tables of values. Then determine the exact value of the limit by using algebra to simplify the function. Finally, plot each function on an appropriate interval to check your result visually.
Recall that our primary motivation for considering limits of functions comes from our interest in studying the rate of change of a function. To that end, we close this section by revisiting our previous work with average and instantaneous velocity and highlighting the role that limits play.
Suppose that we have a moving object whose position at time is given by a function . We know that the average velocity of the object on the time interval is . We define the instantaneous velocity at to be the limit of average velocity as approaches . Note particularly that as , the length of the time interval gets shorter and shorter (while always including ). We will write for the instantaneous velocity at , and thus
For the moving object whose position at time is given by the graph in Figure 1.2.10, answer each of the following questions. Assume that is measured in feet and is measured in seconds.
Figure1.2.10.Plot of the position function in Activity 1.2.4.
Use the graph to estimate the average velocity of the object on each of the following intervals: ,,. Draw each line whose slope represents the average velocity you seek.
How could you use average velocities or slopes of lines to estimate the instantaneous velocity of the object at a fixed time?
Use the graph to estimate the instantaneous velocity of the object when . Should this instantaneous velocity at be greater or less than the average velocity on that you computed in (a)? Why?
Limits enable us to examine trends in function behavior near a specific point. In particular, taking a limit at a given point asks if the function values nearby tend to approach a particular fixed value.
We read , as “the limit of as approaches is ,” which means that we can make the value of as close to as we want by taking sufficiently close (but not equal) to .
To find for a given value of and a known function , we can estimate this value from the graph of , or we can make a table of function values for -values that are closer and closer to . If we want the exact value of the limit, we can work with the function algebraically to understand how different parts of the formula for change as .
We find the instantaneous velocity of a moving object at a fixed time by taking the limit of average velocities of the object over shorter and shorter time intervals containing the time of interest.
(c) Graph the function to see if it is consistent with your answers to parts (a) and (b). By graphing, find an interval for near zero such that the difference between your conjectured limit and the value of the function is less than 0.01. In other words, find a window of height 0.02 such that the graph exits the sides of the window and not the top or bottom. What is the window?
Use a sequence of values of near to estimate the value of , if you think the limit exists. If you think the limit doesn’t exist, explain why.
Use algebra to simplify the expression and hence work to evaluate exactly, if it exists, or to explain how your work shows the limit fails to exist. Discuss how your findings compare to your results in (b).
True or false: . Why?
True or false: . Why? How is this equality connected to your work above with the function ?
Based on all of your work above, construct an accurate, labeled graph of on the interval , and write a sentence that explains what you now know about .
Use a sequence of values near to estimate the value of , if you think the limit exists. If you think the limit doesn’t exist, explain why.
Use algebra to simplify the expression and hence work to evaluate exactly, if it exists, or to explain how your work shows the limit fails to exist. Discuss how your findings compare to your results in (b). (Hint: whenever , but whenever .)
True or false: . Why?
True or false: . Why? How is this equality connected to your work above with the function ?
Based on all of your work above, construct an accurate, labeled graph of on the interval , and write a sentence that explains what you now know about .