copyright © cengage learning. all rights reserved. 3 introduction to the derivative

Copyright © Cengage Learning. All rights reserved.

3 Introduction to theDerivative

Copyright © Cengage Learning. All rights reserved.

3.1 Limits: Numerical and Graphical Viewpoints

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Limits: Numerical and Graphical Viewpoints

Rates of change are calculated by derivatives, but an important part of the definition of the derivative is something called a limit.

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Evaluating Limits Numerically

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Start with a very simple example: Look at the functionf (x) = 2 + x and ask: What happens to f (x) as x approaches 3?

The following table shows the value of f (x) for values of x close to and on either side of 3:

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We have left the entry under 3 blank to emphasize that when calculating the limit of f (x) as x approaches 3, we are not interested in its value when x equals 3.

Notice from the table that the closer x gets to 3 from either side, the closer f (x) gets to 5. We write this as

The limit of f (x), as x approaches 3, equals 5.

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Example 1 – Estimating a Limit Numerically

Use a table to estimate the following limits:

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Example 1(a) – Solution

We cannot simply substitute x = 2, because the function

is not defined at x = 2.

Instead, we use a table of values, with x approaching 2 from both sides.

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We notice that as x approaches 2 from either side, f (x) appears to be approaching 12.

This suggests that the limit is 12, and we write

cont’d

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Example 1(b) – Solution

The function is not defined at x = 0 (nor can it

even be simplified to one which is defined at x = 0).

In the following table, we allow x to approach 0 from both sides:

The table suggests that

cont’d

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Definition of a Limit

If f (x) approaches the number L as x approaches (but is not equal to) a from both sides, then we say that f (x) approaches L as x → a (“x approaches a”) or that the limit of f (x) as x → a is L.

More precisely, we can make f (x) be as close to L as we like by choosing any x sufficiently close to (but not equal to) a on either side. We write

or

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If f (x) fails to approach a single fixed number as x approaches a from both sides, then we say that f (x) has no limit as x → a, or

does not exist.

Quick Example:

As x approaches –2, 3x approaches –6.

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Example 2 – Limits Do Not Always Exist

Do the following limits exist?

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Here is a table of values for with x approaching 0 from both sides.

The table shows that as x gets closer to zero on either side, f (x) gets larger and larger without bound—that is, if you name any number, no matter how large, f (x) will be even larger than that if x is sufficiently close to 0.

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Because f (x) is not approaching any real number, we

conclude that does not exist.

Because f (x) is becoming arbitrarily large, we also say that

diverges to or just

cont’d

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Here is a table of values for with x approaching 0 from both sides.

The table shows that f (x) does not approach the same limit as x approaches 0 from both sides.

There appear to be two different limits: the limit as we approach 0 from the left and the limit as we approach from the right.

cont’d

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We write

read as “the limit as x approaches 0 from the left (or from below) is –1” and

read as “the limit as x approaches 0 from the right (or from above) is 1.”

These are called the one-sided limits of f (x). In order for f to have a two-sided limit, the two one-sided limits must be equal. Because they are not, we conclude thatlimx→0 f (x) does not exist.

cont’d

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Example 2(c) – Solution

Near x = 2, we have the following table of values for

Because f(x) is approaching no (single) real number as

x → 2, we see that does not exist.

cont’d

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Example 2(c) – Solution

Notice also that diverges to as x → 2 from the

positive side (right half of the table) and to as x → 2

from the left (left half of the table).

In other words,

cont’d

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Example 2(d) – Solution

The natural domain of is as f(x) is defined only when Thus we cannot evaluate f(x) if x is to the left of 1.

This means that

does not exist,

and our table looks like this:

cont’d

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Example 2(d) – Solution

The values suggest that

In fact, we can obtain this limit by substituting x = 1 in the formula for f(x).

cont’d

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Estimating Limits Graphically

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Example 4 – Estimating Limits Graphically

The graph of a function f is shown in Figure 1. (We know that the solid dots indicate points on the graph, and the hollow dots indicate points not on the graph.)

From the graph, analyze the following limits.

Figure 1

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Example 4 – Solution

Since we are given only a graph of f, we must analyze these limits graphically.

(a) Imagine that Figure 1 was drawn on a graphing calculator equipped with a trace feature that allows us to

move a cursor along the graph and see the coordinates as we go.

To simulate this, place a pencil point

on the graph to the left of x = –2, and

move it along the curve so that the

x-coordinate approaches –2.

(See Figure 2.)Figure 2

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We evaluate the limit numerically by noting the behavior of the y-coordinates.

However, we can see directly from the graph that they-coordinate approaches 2.

Similarly, if we place our pencil point to the right of x = –2 and move it to the left, the y coordinate will approach 2 from that side as well (Figure 3).

Therefore, as x approaches –2 from either side, f (x) approaches 2, so

Figure 3

cont’d

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(b) This time we move our pencil point toward x = 0. Referring to Figure 4, if we start from the left of x = 0

and approach 0 (by moving right), the y-coordinate approaches –1.

However, if we start from the right of x = 0 and approach 0 (by moving left), the y-coordinate approaches 3.

Thus

and

cont’d

Figure 4

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Because these limits are not equal, we conclude that does not exist.

In this case there is a “break” in the graph at x = 0, and we say that the function is discontinuous at x = 0.

(c) Once more we think about a

pencil point moving along

the graph with the x-coordinate

this time approaching x = 1 from the

left and from the right (Figure 5).

cont’d

Figure 5

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As the x-coordinate of the point approaches 1 from either side, the y-coordinate approaches 1 also.

Therefore,

(d) For this limit, x is supposed

to approach infinity. We think

about a pencil point moving

along the graph further and

further to the right as shown

in Figure 6.

cont’d

Figure 6

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As the x-coordinate gets larger, the y-coordinate also gets larger and larger without bound.

Thus, f (x) diverges to

Similarly,

cont’d

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Evaluating Limits Graphically

To decide whether limx→a f (x) exists and to find its value if it does:

1. Draw the graph of f (x) by hand or with graphing technology.

2. Position your pencil point (or the Trace cursor) on a point

of the graph to the right of x = a.

3. Move the point along the graph toward x = a from the right and read the y-coordinate as you go. The value the

y-coordinate approaches (if any) is the limit limx→a+ f (x).

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4. Repeat Steps 2 and 3, this time starting from a point on the graph to the left of x = a, and approaching x = a along the graph from the left. The value the y-coordinate

approaches (if any) is limx→a– f (x).

5. If the left and right limits both exist and have the same

value L, then limx→a f (x) = L. Otherwise, the limit does

not exist. The value f (a) has no relevance whatsoever.

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6. To evaluate limx→+ f (x), move the pencil point toward

the far right of the graph and estimate the value the

y-coordinate approaches (if any). For limx→– f (x), move

the pencil point toward the far left.

7. If x = a happens to be an endpoint of the domain of f,

then only a one-sided limit is possible at x = a. For

instance, if the domain is (– , 4], then limx→4– f (x)

may exist, but neither limx→4+ f (x) nor limx→4 f (x) exists.

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Application

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Example 6 – Broadband Penetration

Wired broadband penetration in the United States can be modeled by

where t is time in years since 2000.

a. Estimate and interpret the answer.

b. Estimate and interpret the answer.

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Figure 8 shows a plot of P(t) for 0 ≤ t ≤ 20.

Using either the numerical or the graphical approach, we find

Thus, in the long term (as t gets larger and larger), broadband penetration in the United States is expected to approach 30%; that is, the number of installations is expected to approach 30% of the total population.

Figure 8

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The limit here is

(Notice that in this case, we can simply put t = 0 to evaluate this limit.) Thus, the closer t gets to 0 (representing 2000) from the right, the closer P(t) gets to 2.21%, meaning that, in 2000, broadband penetration was about 2.2% of thepopulation.

cont’d

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