4. 4
Infinite Limits
As x 0, f(x) gets bigger and bigger without bound
To indicate this kind of behavior we use the notation
5. 5
Infinite Limits
Again, the symbol is not a number, but the expression
limx®a 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
6. 6
Infinite Limits
Similarly f (x) decreases without bound as x approaches a:
7. 7
Infinite Limits
Similar 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.
11. 11
Vertical Asymptotes
For instance, the y-axis is a vertical asymptote of the curve
y = 1/x2 because limx®0 (1/x2) = .
Note: Asymptotes are lines – described by equations
e.g., x = 0
12. Example 1 – Evaluating One-sided Infinite Limits
12
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
13. 13
Example 1 – Solution
Likewise, 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
14. 14
Example 1 – Solution
The line x = 3 is a vertical
asymptote.
cont’d
15. 15
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
16. 16
Vertical Asymptotes of tan(x)
and so the line x = p/2 is a vertical asymptote.
In fact, the lines x = (2n + 1)p/2,
n an integer, are all vertical
asymptotes of y = tan x.
Figure 7 y = tan x
20. 20
Limits at Infinity
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.
21. 21
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
22. 22
Limits at Infinity
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
23. 23
Limits at Infinity
In general, we use the notation
to indicate that the values of f (x) approach L as x becomes
larger and larger.
24. 24
Limits at Infinity
The 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”
25. 25
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
26. 26
Limits at Infinity
Referring to Figure 8, we see that for numerically large
negative values of x, the values of f (x) are close to 1.
Figure 8
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
27. 27
Limits at Infinity
In 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
29. 29
Horizontal Asymptote - Graphically
For instance, the curve illustrated in Figure 8
Figure 8
has the line y = 1 as a horizontal asymptote because
30. 2 Horizontal Asymptotes
The curve y = f (x) sketched in Figure 11 has both y = –1 and
y = 2 as horizontal asymptotes because
30
Figure 11
31. 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.
31
Figure 12
32. Example 3 – Solution
We see that the values of f (x) become large as x ® –1 from
both sides, so
32
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.
33. 33
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
35. 35
Limits at Infinity
Most 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.
36. 36
Limits at Infinitely - Algebraically
Divide each term by highest power of x in denominator:
37. 37
Limits at Infinity
The 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
39. 39
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:
40. 40
Infinite Limits at Infinity
From 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
