10
Introduction to the
Derivative
Copyright © Cengage Learning. All rights reserved.
10.1
Limits: Numerical and Graphical Approaches
Copyright © Cengage Learning. All rights reserved.
Evaluating Limits Numerically
3
Evaluating Limits Numerically
Rates of change are calculated by derivatives, but an
important part of the definition of the derivative is
something called a limit.
Start with a very simple example: Look at the function
f(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:
4
Evaluating Limits Numerically
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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Example 1(a) – Solution
cont’d
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
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Example 1(b) – Solution
cont’d
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
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Evaluating Limits Numerically
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
10
Evaluating Limits Numerically
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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Example 2(a) – Solution
Here is a table of values for
from both sides.
with x approaching 0
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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Example 2(a) – Solution
cont’d
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
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Example 2(b) – Solution
Here is a table of values for
from both sides.
cont’d
with x approaching 0
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.
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Example 2(b) – Solution
cont’d
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 that
limx→0 f(x) does not exist.
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Example 2(c) – Solution
cont’d
Near x = 2, we have the following table of values for
Because
is approaching no (single) real number as
x → 2, we see that
does not exist.
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Example 2(c) – Solution
cont’d
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,
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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. (Recall 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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Example 4 – Solution
cont’d
We evaluate the limit numerically by noting the behavior
of the y-coordinates.
However, we can see directly from the graph that the
y-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
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Example 4 – Solution
cont’d
(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
Figure 4
and
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Example 4 – Solution
cont’d
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).
Figure 5
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Example 4 – Solution
cont’d
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.
Figure 6
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Example 4 – Solution
cont’d
As the x-coordinate gets larger, the y-coordinate also
gets larger and larger without bound.
Thus, f(x) diverges to
Similarly,
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Estimating Limits Graphically
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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Estimating Limits Graphically
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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Estimating Limits Graphically
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)
can be computed, but not limx→4+ f(x). In this case, we
have said that limx→4 f(x) is just the one-sided limit
limx→4– f(x).
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Application
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Example 6 – Broadband
The percentage of U.S. Internet-connected households that
have broadband connections can be modeled by
where t is time in years since 2000.
a. Estimate
b. Estimate
and interpret the answer.
and interpret the answer.
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Example 6(a) – Solution
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
Figure 8
larger and larger), the percentage
of U.S. Internet connected households
that have broadband is expected to approach 100%.
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Example 6(b) – Solution
cont’d
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) the closer
P(t) gets to 8.475%, meaning that, in 2000, about 8.5% of
Internet-connected households had broadband.
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