Download (6) Prove that the equation x

10. LIMITS AT INFINITY
29
gap in the graph of the function, contradicting the requirement that
the function be continuous. This is the content of the next theorem.
Theorem 6 (Intermediate Value Theorem). Let f be a function
that is continuous on the interval [a, b]. Then, if f (a) = y1 and f (b) =
y2 and y is a real number that is between y1 and y2 , then there exists
x ∈ [a, b] such that f (x) = y.
In other words, f takes all values between y1 and y2 on the interval
[a, b].
Example 17. There is a real number x in the interval [0, 1] such
that x + ex = 2.
Solution: Let f (x) = x+ex . Then f is continuous everywhere, f (0) =
1, and f (1) = 1 + e > 3.71. So, by the intermediate value theorem,
we get that f takes all values between 1 and 1 + e on that interval,
including y = 2.
2
9.6. Exercises.
(1)
(2)
(3)
(4)
(5)
Where is tan x continuous?
Where is 1/x not continuous?
continuous?
Where is 5x23x+2
−6x+1
2
Where is sin(x ) continuous?
Prove that the equation x5 −x−1 = 0 has a root in the interval
(−1, 2).
(6) Prove that the equation x3 − 3x − 1 = 0 has at least two roots
in the interval (−1, 2).
10. Limits at Infinity
10.1. Finite Limits at Infinity. In Section 7, we defined what it meant
for a function to have a limit L at a real number a. In this section, we
extend that definition and define what it means for a function to have
a limit L at ∞ or at −∞.
Definition 14. Let f : R → R be a function that is defined on
some interval (b, ∞). We say that the limit of f at ∞ is the real number
L if the values of f (x) get arbitrarily close to L and stay arbitrarily close
to L when x is suitably large.
The fact that the limit of f at ∞ is L is expressed by the notation
lim f (x) = L.
x→∞
This definition follows the idea of the definition of limits at finite
points. Indeed, in order for limx→∞ f (x) = L to hold, we require that
30
2. LIMITS AND DERIVATIVES
the values of f (x) get arbitrarily close to L and stay arbitrarily close
to L if x is large enough. Here “x is large enough” means that x is
in a suitably selected neighborhood of ∞, in other words, in an open
interval (c, ∞). Recall that this is analogous to what we required in the
finite case. There we said that limx→a f (x) = L if f (x) got arbitrarily
close to L and stayed arbitrarily close to L once x was suitably close
to a, that is, when x was in a suitably selected neighborhood of a.
Example 18. Let f (x) = 1/x. Then
lim f (x) = 0.
x→∞
Solution: If we want the value of f (x) to be closer than to 0, all we
have to do is to select x such that x > 1/ holds. Once x gets past 1/,
the values of f (x) will stay between 0 and .
2
The definition of limits at −∞ is what the reader probably expects.
Definition 15. Let f : R → R be a function defined on some
interval (−∞, b). We say that the limit of f at −∞ is the real number
L if the values of f (x) get arbitrarily close to L and stay arbitrarily
close to L when x is a negative number with a suitably large absolute
value.
The fact that the limit of f at −∞ is L is expressed by the notation
lim f (x) = L.
x→−∞
Example 19. Let f (x) = 1/x2 . Then
lim f (x) = 0.
x→−∞
Solution: If we want to get f (x) closer√
than to 0 and keep it there,
it suffices to choose x such that x < −1/ . Then x2 > 1/, and hence
2
f (x) = 1/x2 < .
10.1.1. The Formal Definition of Limits at Infinity. The formal definition of limits at infinity is very similar to that of limits at finite points.
The only difference is in the formal description of what it means to be
in a neighborhood of infinity versus what it means to be in a neighborhood of a real number.
Definition 16. Let f : R → R be a function defined on some
interval (b, ∞). We say that limx→∞ f (x) = L if, for all positive real
numbers , there exists a positive real number N such that if x > N ,
then |f (x) − L| < .
10. LIMITS AT INFINITY
31
The formal definition of limits at negative infinity is analogous. The
only difference is again in the formal description of what it means for x
to be in a neighborhood of −∞. It means to be in an interval (−∞, c).
Definition 17. Let f : R → R be a function defined on some
interval (−∞, b). We say that limx→−∞ f (x) = L if, for all positive real
numbers , there exists a negative real number N such that if x < N ,
then |f (x) − L| < .
10.1.2. The Graphical Meaning of a Finite Limit at Infinity. If a func-
tion f has limit L at ∞ or −∞, then the graph of the function will
approach the horizontal line y = L at that infinity. The graph may or
may not actually touch that line or even become that line. The line
y = L is called a horizontal asymptote of the graph of y = f (x) when
limx→∞ f (x) = L or limx→−∞ f (x) = L holds.
10.2. Infinite Limits at Infinity. It can happen that the limit of a function
at ∞ is not a real number but rather ∞ or −∞.
Definition 18. Let f : R → R be a function defined on some
interval (b, ∞). We say that the limit of f at ∞ is ∞, denoted by
lim f (x) = ∞,
x→∞
if f (x) gets arbitrarily large and stays arbitrarily large if x gets sufficiently large.
Example 20. Let f (x) = ex . Then limx→∞ f (x) = ∞.
Solution: In order to get f (x) to be larger than some given positive
real number M , it suffices to choose x > ln M .
2
The following notation is defined in an analogous way:
(I) limx→∞ f (x) = −∞.
(II) limx→−∞ g(x) = ∞.
(III) limx→−∞ h(x) = −∞.
Each of these definitions refers to a fact that the values of a function
get arbitrarily far away from 0 and stay arbitrarily far away from 0 (in
the appropriate direction) if x gets sufficiently far away from 0 (in the
appropriate direction). The reader should test his or her understanding of these concepts by verifying that limx→∞ 1 − x = −∞, while
limx→−∞ x2 = ∞, and limx→−∞ x3 = −∞.
32
2. LIMITS AND DERIVATIVES
10.2.1. The Formal Definition of Infinite Limits at Infinity. By now, the
formal definition of infinite limits at infinity probably does not come
as a surprise. We are providing a formal definition for one of the four
possible scenarios that can occur due to changes in sign. The other
three cases are analogous.
Definition 19. Let f : R → R be a function defined on some
interval (b, ∞). We say that limx→∞ f (x) = ∞ if, for all positive real
numbers M , there exists a positive real number N such that if x > N ,
then f (x) > M .
10.3. Computing Limits at Infinity. The limit laws that we learned for
limits at finite points stay true for limits at infinity as well, provided,
of course, that they make sense. Here are a few examples.
Example 21. We have
x+3
= 1.
x→∞ x − 4
lim
It would be wrong to argue as follows: “The numerator is the function f (x) = x + 3, and the denominator is the function g(x) = x − 4.
At ∞, they both have limit ∞, so, by the limit law for quotients, the
limit of their quotient is 1.”
The problem with this argument is that ∞ is not a number. So
∞/∞ is not defined. It is possible for f and g both to have limit ∞
at ∞, and for f /g to have limits c at ∞, for any given real number c.
Indeed, let f (x) = cx and let g(x) = x.
Instead, we can solve Example 21 as follows.
Solution:
x+3
(x − 4) + 7
= lim
x→∞ x − 4
x→∞
x−4 7
= lim 1 +
x→∞
x−4
7
= 1 + lim
x→∞ x − 4
=1+0
= 1.
lim
2
We would like to point out other pitfalls when dealing with the
application of limit laws and infinite limits. The following expressions
are not defined:
10. LIMITS AT INFINITY
33
(I) ∞ + (−∞)
(II) ∞ · 0 and −∞ · 0
(III) 1∞ and 1−∞
The following theorem is very useful when dealing with limits at ∞.
Theorem 7. Let r be a positive rational number. Then
1
= 0.
x→∞ xr
lim
If r is an integer, then this statement follows from the fact that
limx→∞ 1/x = 0 by applying limit law III (for products) r times. If
r = p/q, where p and q are positive integers, then we can first
prove
√
q
p
p/q
p
the theorem for x , and then, using the root law, for x = x .
Many limits can be computed with the help of this theorem.
Example 22. We have
x2 + 3x + 1
= 0.
lim
x→∞
x3
Solution: We have
x2 + 3x + 1
x2 3x
1
=
+ 3 + 3,
3
3
x
x
x
x
and each of the three summands has limit 0 at ∞ by the preceding
theorem. Hence, by the limit law for sums, so does their sum.
2
Note that the limit would not change if we changed the denominator
from x3 to x3 +3x2 +4x+5. This would have decreased the value of our
function, but would have still kept it positive. Hence, by the squeeze
principle, we can then conclude that
x2 + 3x + 1
= 0.
x→∞ x3 + 3x2 + 4x + 5
lim
10.4. Exercises.
(1)
(2)
(3)
(4)
(5)
Find limx→∞ xx+1
2 +4 .
3x2 +4x+1
Find limx→∞ x2 +4 .
3
.
Find limx→∞ xx2+2x
+4
3x2 +4x+1
Find limx→−∞ x2 +4 .
Let R(x) = p(x)/q(x) be a rational function. Explain how
limx→∞ R(x) depends on p(x) and q(x).