1. Copyright © Cengage Learning. All rights reserved.
Limits: A Preview of Calculus
Copyright © Cengage Learning. All rights reserved.

2. 13.1 Finding Limits Numerically and Graphically
Copyright © Cengage Learning. All rights reserved.

3. Objectives Definition of Limit
Estimating Limits Numerically and Graphically
Limits That Fail to Exist
One-Sided Limits

4. Definition of Limit

5. Definition of Limit In general, we use the following notation.
Roughly speaking, this says that the values of f (x) get closer and closer to the number L as x gets closer and closer to the number a (from either side of a) but x ≠ a.

6. Definition of Limit An alternative notation for limx  a f (x) = L is
f (x)  L as x  a
which is usually read “f (x) approaches L as x approaches a.”
Notice the phrase “but x ≠ a” in the definition of limit. This means that in finding the limit of f (x) as x approaches a, we never consider x = a. In fact, f (x) need not even be defined when x = a. The only thing that matters is how f is defined near a.

7. Definition of Limit
Figure 2 shows the graphs of three functions. Note that in part (c), f (a) is not defined, and in part (b), f (a) ≠ L. But in each case, regardless of what happens at a, limxa f (x) = L.
(a)
(b)
(c)
f (x) = L in all three cases
Figure 2

8. Estimating Limits Numerically and Graphically

9. Example 1 – Estimating Limits Numerically and Graphically
Estimate the value of the following limit by making a table of values. Check your work with a graph.
Solution: Notice that the function f (x) = (x – 1)/(x2 – 1) is not defined when x = 1, but this doesn’t matter because the definition of limxa f (x) says that we consider values of x that are close to a but not equal to a.

10. Example 1 – Solution
cont’d
The following tables give values of f (x) (rounded to six decimal places) for values of x that approach 1 (but are not equal to 1).
On the basis of the values in the two tables, we make the guess that

11. Example 1 – Solution
cont’d
As a graphical verification we use a graphing device to produce Figure 3. We see that when x is close to 1, y is close to 0.5. If we use the and features to get a closer look, as in Figure 4, we notice that as x gets closer and closer to 1, y becomes closer and closer to 0.5. This reinforces our conclusion.
Figure 3
Figure 4

12. Limits That Fail to Exist

13. Limits That Fail to Exist
Functions do not necessarily approach a finite value at every point. In other words, it’s possible for a limit not to exist.
The next example illustrates ways in which this can happen.

14. Example 3 – A Limit That Fails to Exist (A Function with a Jump)
The Heaviside function H is defined by
[This function, named after the electrical engineer Oliver Heaviside (1850–1925), can be used to describe an electric current that is switched on at time t = 0.] Its graph is shown in Figure 6. Notice the “jump” in the graph at x = 0.
Figure 6

15. Example 3 – A Limit That Fails to Exist (A Function with a Jump)
cont’d
As t approaches 0 from the left, H (t) approaches 0. As t approaches 0 from the right, H (t) approaches 1.
There is no single number that H (t) approaches as t approaches 0. Therefore, limt0 H (t) does not exist.

16. One-Sided Limits

17. One-Sided Limits
We noticed in Example 3 that H(t) approaches 0 as t approaches 0 from the left H (t) and approaches 1 as t approaches 0 from the right.
We indicate this situation symbolically by writing
and
The symbol “t  0–” indicates that we consider only values of t that are less than 0. Likewise, “t  0+” indicates that we consider only values of t that are greater than 0.

18. One-Sided Limits
Notice that this definition differs from the definition of a two-sided limit only in that we require x to be less than a.

19. One-Sided Limits
Similarly, if we require that x be greater than a, we get “the right-hand limit of f (x) as x approaches a is equal to L,” and we write
Thus the symbol “x  a+” means that we consider only x > a. These definitions are illustrated in Figure 9.
(a) f (x) = L
(b) f (x) = L
Figure 9

20. One-Sided Limits
By comparing the definitions of two-sided and one-sided limits, we see that the following is true.
Thus if the left-hand and right-hand limits are different, the (two-sided) limit does not exist. We use this fact in the next example.

21. Example 6 – Limits from a Graph
The graph of a function g is shown in Figure 10. Use it to state the values (if they exist) of the following:
(a)
(b)
Figure 10

22. Example 6(a) – Solution
From the graph we see that the values of g (x) approach 3 as x approaches 2 from the left, but they approach 1 as x approaches 2 from the right.
Therefore
and
Since the left- and right-hand limits are different, we conclude that limx2 g (x) does not exist.

23. Example 6(b) – Solution The graph also shows that and
cont’d
The graph also shows that
and
This time the left- and right-hand limits are the same, so we have
Despite this fact, notice that g (5) ≠ 2.