Limits Involving Infinity2.5
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Infinite Limits
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Infinite Limits
does not exist
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Infinite LimitsAs x 0, f(x) gets bigger and bigger without bound
To indicate this kind of behavior we use the notation
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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 f (x) decreases without bound as x approaches a:
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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 Limits – One-Sided Limits
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Infinite Limits - Examples
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Infinite Limits - Example
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Vertical Asymptotes
For instance, the y-axis is a vertical asymptote of the curve
y = 1/x2 because limx0 (1/x2) = .
Note: Asymptotes are lines – described by equations
e.g., x = 0
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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 – Solution
The line x = 3 is a vertical asymptote.
cont’d
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Vertical Asymptote of ln(x)Two familiar functions whose graphs have vertical asymptotes are y = ln x and y = tan x.
and so the line x = 0 (the y-axis)
is a vertical asymptote.
Figure 6
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Vertical Asymptotes of tan(x)
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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Vertical Asymptotes - Examples
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Vertical Asymptote - Example
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Limits at Infinity
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Limits at InfinityHere 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.
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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”
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Limits at Infinity
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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Horizontal Asymptote
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Horizontal Asymptote - GraphicallyFor instance, the curve illustrated in Figure 8
has the line y = 1 as a horizontal asymptote because
Figure 8
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2 Horizontal AsymptotesThe 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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Limit of 1/x as x∞x 1/x
1 1
10 0.1
20 0.05
100 0.01
1000 0.001
10,000 0.0001
-1 -1
-10 -0.1
-20 -0.05
100 -0.01
1000 -0.001
10,000 -0.0001
x
y
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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 Infinitely - AlgebraicallyDivide each term by highest power of x in denominator:
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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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Limits at Infinity Examples