Copyright © Cengage Learning. All rights reserved.

Limits: A Preview of Calculus

Copyright © Cengage Learning. All rights reserved.

13.1 Finding Limits Numericallyand Graphically

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Objectives

► Definition of Limit

► Estimating Limits Numerically and Graphically

► Limits That Fail to Exist

► One-Sided Limits

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Definition of Limit

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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.

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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.

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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, limxa f (x) = L.

(c)(a) (b)

f (x) = L in all three cases

Figure 2

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Estimating Limits Numerically and Graphically

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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 limxa f (x) says that we consider values of x that are close to a but not equal to a.

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Example 1 – Solution

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

cont’d

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Example 1 – Solution

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

cont’d

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Limits That Fail to Exist

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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.

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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

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Example 3 – A Limit That Fails to Exist (A Function with a Jump)

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, limt0 H (t) does not exist.

cont’d

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One-Sided Limits

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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.

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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.

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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.

Figure 9

(a) f (x) = L (b) f (x) = L

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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.

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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

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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 limx2 g (x) does not exist.

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Example 6(b) – Solution

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.

cont’d