Limits at Infinity; Horizontal Asymptotes
In this section 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; Horizontal Asymptotes
The table gives values of this function correct to six decimal places, and the graph of f has been drawn by a computer in Figure 1.
Figure 1
Limits at Infinity; Horizontal Asymptotes
As 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 Infinity; Horizontal Asymptotes
In general, we use the notation
to indicate that the values of f(x) approach L as x becomes larger and larger.
Limits at Infinity; Horizontal Asymptotes
Another notation for is f(x) → L as x →
Geometric illustrations of Definition 1 are shown in Figure 2.
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Limits at Infinity; Horizontal Asymptotes
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.
Limits at Infinity; Horizontal Asymptotes
Referring back to Figure 1, 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 1
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Limits at Infinity; Horizontal Asymptotes
The general definition is as follows.
Limits at Infinity; Horizontal Asymptotes
Definition 2 is illustrated in Figure 3. Notice that the graph approaches the line y = L as we look to the far left of each graph.
Examples illustrating
Figure 3
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Limits at Infinity; Horizontal Asymptotes
Example 2
Find and Solution:
Observe that when x is large, 1/x is small. For instance,
In fact, by taking x large enough, we can make 1/x as close to 0 as we please.
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Example 2 – Solution
Therefore, according to Definition 1, we have
= 0
Similar reasoning shows that when x is large negative, 1/x is small negative, so we also have
= 0
cont’d
Example 2 – Solution
It follows that the line y = 0 (the x-axis) is a horizontal asymptote of the curve y = 1/x. (This is an equilateral hyperbola; see Figure 6.)
Figure 6
cont’d
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Limits at Infinity; Horizontal Asymptotes
Example 3
Evaluate and indicate which properties of limits are used at each stage.
Solution:
As x becomes large, both numerator and denominator
become large, so it isn’t obvious what happens to their ratio.
We need to do some preliminary algebra.
To evaluate the limit at infinity of any rational function, we first divide both the numerator and denominator by the highest power of x that occurs in the denominator. (We may assume that x ≠ 0, since we are interested only in
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Example 3 – Solution
In this case the highest power of x in the denominator is x2, so we have
cont’d
Example 3 – Solution
cont’d(by Limit Law 5)
(by 1, 2, and 3)
(by 7 and Theorem 4)
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Example 3 – Solution
A similar calculation shows that the limit as x → – is also
Figure 7 illustrates the results of these calculations by showing how the graph of the given
rational function approaches the horizontal asymptote
y = = 0.6.
cont’d
Figure 7
Example 4
Find the horizontal and vertical asymptotes of the graph of the function
Solution:
Dividing both numerator and denominator by x and using the properties of limits, we have
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Example 4 – Solution
cont’d(since = x for x > 0)
Example 4 – Solution
Therefore the line y = is a horizontal asymptote of the graph of f.
In computing the limit as x → – , we must remember that for x < 0, we have = | x| = –x.
cont’d
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Example 4 – Solution
So when we divide the numerator by x, for x < 0 we get
cont’d
Example 4 – Solution
Therefore
cont’d
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Example 4 – Solution
Thus the line y = – is also a horizontal asymptote.
A vertical asymptote is likely to occur when the denominator, 3x – 5, is 0, that is, when
If x is close to and x > , then the denominator is close to 0 and 3x – 5 is positive. The numerator is always positive, so f(x) is positive.
Therefore
cont’d
Example 4 – Solution
If x is close to but x < , then 3x – 5 < 0 and so f(x) is large negative. Thus
The vertical asymptote is x = . All three asymptotes are shown in Figure 8.
cont’d
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Infinite Limits at Infinity
Infinite Limits at Infinity
The 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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Example 8
Find and Solution:
When x becomes large, x3 also becomes large. For instance,
In fact, we can make x3 as big as we like by requiring x to be large enough. Therefore we can write
Example 8 – Solution
Similarly, when x is large negative, so is x3. Thus
These limit statements can also be seen from the graph of y = x3 in Figure 10.
cont’d
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Precise Definitions
Precise Definitions
Definition 1 can be stated precisely as follows.
In words, this says that the values of f(x) can be made arbitrarily close to L (within a distance ε, where ε is any positive number) by requiring x to be sufficiently large (larger than N, where N depends on ε).
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Precise Definitions
Graphically it says that by keeping x large enough (larger than some number N) we can make the graph of f lie
between the given horizontal lines y = L – ε and y = L + ε as in Figure 12.
Figure 12
Precise Definitions
This must be true no matter how small we choose ε.
Figure 13 shows that if a smaller value of ε is chosen, then a larger value of N may be required.
Figure 13
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Precise Definitions
Example 13
Use Definition 5 to prove that = 0.
Solution:
Given ε > 0, we want to find N such that if x > N then
In computing the limit we may assume that x > 0.
Then
1/x < ε x > 1/ε.
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Example 13 – Solution
Let’s choose N = 1/ε. So if then Therefore, by Definition 5,
= 0
cont’d
Example 13 – Solution
Figure 16 illustrates the proof by showing some values of ε and the corresponding values of N.
cont’d
Figure 16
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Precise Definitions
Finally we note that an infinite limit at infinity can be defined as follows. The geometric illustration is given in Figure 17.
Figure 17
Similar definitions apply when the symbol is replaced by – .
