Limits and Derivatives 2. Limits Involving Infinity 2.5.

Limits and Derivatives2

Limits Involving Infinity2.5

3

Limits Involving Infinity

In this section we investigate the global behavior of

functions and, in particular, whether their graphs approach

asymptotes, vertical or horizontal.

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Infinite Limits

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Infinite LimitsWe have concluded that

does not exist

by observing, from the table of values and the graph of y = 1/x2 in Figure 1, that the values of 1/x2 can be made arbitrarily large by taking x close enough to 0.

Figure 1

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Infinite LimitsThus the values of f (x) do not approach a number, so

limx0 (1/x2) does not exist.

To indicate this kind of behavior we use the notation

In general, we write symbolically

to indicate that the values of f (x) become larger and larger (or “increase without bound”) as x approaches a.

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Infinite Limits

Another notation for limxa f (x) = is

f (x) as x a

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Infinite LimitsAgain, the symbol is not a number, but the expression

limxa f (x) = is often read as

“the limit of f (x), as x approaches a, is infinity”

or “f (x) becomes infinite as x approaches a”

or “f (x) increases without bound as x approaches a”

This definition is illustrated

graphically in Figure 2.

Figure 2

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Infinite LimitsSimilarly, as shown in Figure 3,

means that the values of f (x) are as large negative as we like for all values of x that are sufficiently close to a, but not equal to a.

Figure 3

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Infinite LimitsThe symbol limxa f (x) = can be read as “the limit of f (x),

as x approaches a, is negative infinity” or “f (x) decreases

without bound as x approaches a.”

As an example we have

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Infinite LimitsSimilar definitions can be given for the one-sided infinite limits

remembering that “x a–” means that we consider only values of x that are less than a, and similarly “x a+” means that we consider only x > a.

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Infinite LimitsIllustrations of these four cases are given in Figure 4.

Figure 4

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Infinite Limits

For instance, the y-axis is a vertical asymptote of the curve

y = 1/x2 because limx0 (1/x2) = .

In Figure 4 the line x = a is a vertical asymptote in each of the four cases shown.

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Example 1 – Evaluating One-sided Infinite Limits

Find and

Solution:

If x is close to 3 but larger than 3, then the denominator

x – 3 is a small positive number and 2x is close to 6.

So the quotient 2x/(x – 3) is a large positive number.

Thus, intuitively, we see that

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Example 1 – SolutionLikewise, if x is close to 3 but smaller than 3, then x – 3 is a small negative number but 2x is still a positive number

(close to 6).

So 2x/(x – 3) is a numerically large negative number.

Thus

cont’d

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Example 1 – SolutionThe graph of the curve y = 2x/(x – 3) is given in Figure 5.

The line x = 3 is a vertical asymptote.

cont’d

Figure 5

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Infinite LimitsTwo familiar functions whose graphs have vertical asymptotes are y = ln x and y = tan x.

From Figure 6 we see that

and so the line x = 0 (the y-axis)

is a vertical asymptote.

In fact, the same is true for

y = loga x provided that a > 1.Figure 6

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Infinite LimitsFigure 7 shows that

and so the line x = /2 is a vertical asymptote.

In fact, the lines x = (2n + 1)/2,

n an integer, are all vertical

asymptotes of y = tan x.

Figure 7 y = tan x

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Limits at Infinity

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Limits at InfinityIn computing infinite limits, we let x approach a number and the result was that the values of y became arbitrarily large (positive or negative).

Here we let x become arbitrarily large (positive or negative) and see what happens to y.

Let’s begin by investigating the behavior of the function f defined by

as x becomes large.

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Limits at Infinity

The table at the right gives values

of this function correct to six decimal

places, and the graph of f has been

drawn by a computer in Figure 8.

Figure 8

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Limits at InfinityAs x grows larger and larger you can see that the values of

f (x) get closer and closer to 1.

In fact, it seems that we can make the values of f (x) as close as we like to 1 by taking x sufficiently large.

This situation is expressed symbolically by writing

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Limits at InfinityIn general, we use the notation

to indicate that the values of f (x) approach L as x becomes larger and larger.

Another notation for limx f (x) = L is

f (x) L as x

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Limits at InfinityThe symbol does not represent a number.

Nonetheless, the expression is often read as

“the limit of f (x), as x approaches infinity, is L”

or “the limit of f (x), as x becomes infinite, is L”

or “the limit of f (x), as x increases without bound, is L”

The meaning of such phrases is given by Definition 4.

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Limits at Infinity

Geometric illustrations of Definition 4 are shown in Figure 9.

Notice that there are many ways for the graph of f to

approach the line y = L (which is called a horizontal

asymptote) as we look to the far right of each graph.

Figure 9 Examples illustrating

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Limits at InfinityReferring to Figure 8, we see that for numerically large negative values of x, the values of f (x) are close to 1.

By letting x decrease through negative values without bound, we can make f (x) as close to 1 as we like.

This is expressed by writing

Figure 8

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Limits at InfinityIn general, as shown in Figure 10, the notation

means that the values of f (x) can be made arbitrarily close to

L by taking x sufficiently large negative.

Figure 10 Examples illustrating

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Limits at InfinityAgain, the symbol does not represent a number, but the

expression is often read as

“the limit of f (x), as x approaches negative infinity, is L”

Notice in Figure 10 that the graph approaches the line y = L

as we look to the far left of each graph.

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Limits at InfinityFor instance, the curve illustrated in Figure 8

has the line y = 1 as a horizontal asymptote because

Figure 8

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Limits at InfinityThe curve y = f (x) sketched in Figure 11 has both y = –1 and y = 2 as horizontal asymptotes because

Figure 11

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Example 3 – Infinite Limits and Asymptotes from a Graph

Find the infinite limits, limits at infinity, and asymptotes for the function f whose graph is shown in Figure 12.

Figure 12

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Example 3 – Solution

We see that the values of f (x) become large as x –1 from

both sides, so

Notice that f (x) becomes large negative as x approaches 2

from the left, but large positive as x approaches 2 from the

right. So

Thus both of the lines x = –1 and x = 2 are vertical asymptotes.

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Example 3 – Solution As x becomes large, it appears that f (x) approaches 4.

But as x decreases through negative values, f (x) approaches 2. So

This means that both y = 4 and y = 2 are horizontal asymptotes.

cont’d

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Limits at InfinityMost of the Limit Laws hold for limits at infinity. It can be

proved that the Limit Laws are also valid if “x a” is replaced by “x ” or “x .”

In particular, we obtain the following important rule for

calculating limits.

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Limits at InfinityThe graph of the natural exponential function y = ex has the line y = 0 (the x-axis) as a horizontal asymptote. (The same is true of any exponential function with base a > 1.)

In fact, from the graph in Figure 16 and the corresponding table of values, we see that

Notice that the values of

ex approach 0 very rapidly.Figure 16

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Infinite Limits at Infinity

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Infinite Limits at InfinityThe notation

is used to indicate that the values of f (x) become large as x becomes large.

Similar meanings are attached to the following symbols:

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Infinite Limits at InfinityFrom Figures 16 and 17 we see that

but, as Figure 18 demonstrates, y = ex becomes large as

x at a much faster rate than y = x3.

Figure 16 Figure 17 Figure 18

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Example 9 – Finding an Infinite Limit at Infinity

Find

Solution:

It would be wrong to write

The Limit Laws can’t be applied to infinite limits because

is not a number ( can’t be defined).

However, we can write

because both x and x – 1 become arbitrarily large.