LIMITS AT INFINITY

INTRODUCTION
In the previous section we saw limits that were infinity and it’s now time to take a look at limits at
infinity. By limits at infinity we mean one of the following two limits.

In other
words, we are going to be looking at what happens to a function if we let x get very large in
either the positive or negative sense. Also, as well soon see, these limits may also have infinity
as a value. Limits at infinity are used to describe the behavior of functions as the independent
variable increases or decreases without bound.
OBJECTIVES

• Be able to determine limits at infinity

• Use algebraic techniques to help with indeterminate forms of ±∞ ±∞ and ∞ − ∞
• Use substitutions to evaluate limits of compositions of functions.
• To be able to calculate limits at infinity

ASSESSMENT OF PRIOR KNOWLEDGE: Let’s Review!

Direction: Determine of what is being asked.

1. What is Infinity?
2. Symbol of Infinity.
3. What are the limits of infinity?

DISCUSSION
Limits at Infinity
When graphing a function, we are interested in what happens the values of the function as x
becomes very large in absolute value. For example, if f(x) = 1/x then as x becomes very large
and positive, the values of f(x) approach zero.
f(100) = f(1,000) = f(10,000) = f(1,000,000) =
f(−1,000) = f(−10,000) = f(−1,000,000) =
f(−100) =
We say
lim 1/x = 0 x→∞ and lim 1/x = 0.
x→−∞
Definition Let f be a function defined on some interval (a,∞). Then
lim f(x) = L x→∞
if the values of f(x) can be made arbitrarily close to L by taking x sufficiently large or equivalently if
for any number , there is a number M so that for all .
If f is defined on an interval (−∞,a), then we say

if the values of f(x) can be made arbitrarily close to L by taking x sufficiently large and negative or
equivalently if for any number , there is a number N so that for all .

Note The symbol ∞ here does not represent a number, rather the symbol limx→∞ means the limit
as x becomes increasingly large.
Example Consider the graph of the function shown below. Judging from the graph, find are the
limits

x3 – 2
g(x) = 3
x +1

We can see from the above graph that if limx→∞ f(x) = L, then the graph get closer and closer to
the line y = L as x approaches infinity.

Definition The line y = L is called a horizontal asymptote of the curve y = f(x) if either:
lim f(x) = L or lim f(x) = L.
x→∞ x→−∞

Example

What are the horizontal asymptotes of the graph of shown above?

We saw above that
 =0 and lim .
Example Find the following limits

and lim .
Most of the usual limit laws hold for infinite limits with a replaced by ∞ or −∞. The laws are listed
below for reference :
Suppose that c is a constant and the limits
lim f(x) and lim g(x) x→a x→a
exist (meaning they are finite numbers). Then
1. limx→a[f(x) + g(x)] = limx→a f(x) + limx→a g(x) ; (the limit of a sum is the sum of the
limits).
2. limx→a[f(x) − g(x)] = limx→a f(x) − limx→a g(x) ; (the limit of a difference is the
difference of the limits).
3. limx→a[cf(x)] = climx→a f(x);
(the limit of a constant times a function is the constant times the limit of the function).
4. limx→a[f(x)g(x)] = limx→a f(x) · limx→a g(x); (The limit of a product is the product of
the limits).

5. if limx→a g(x) 6= 0;
(the limit of a quotient is the quotient of the limits provided that the limit of the denominator is not
0)

6. , where n is a positive integer (we see this using

rule 4 repeatedly).
7. limx→a c = c, where c is a constant ( easy to prove and easy to see from the
graph, y = c).
8. limx→a x = a, (not difficult to prove from the definition and easy to see from the
graph, y = x)

) assuming that the limx→a f(x) > 0 if n is even.

Using 9, 6 and 10, we get;
Theorem If r > 0 is a rational number, then

.
If r > 0 is a rational number such that xr is defined for all x, then

Method For Rational Functions

We can use the above theorem to evaluate limits of rational functions at ∞ and −∞. We divide
both the numerator and denominator by the highest power of x in the denominator. Example
Evaluate

and lim .
Find the vertical and horizontal asymptotes of the graph of

Definition Let f be a function defined on some interval (a,∞). Then we say

lim f(x) = ∞ x→∞
if the values of f(x) can be made arbitrarily large by taking x sufficiently large or equivalently if for any
positive integer N, there is a number M so that for all x > M, f(x) > N. We give similar meaning to the
statements
lim f(x) = −∞, lim f(x) = ∞ lim f(x) = −∞.
x→∞ x→−∞ x→−∞

We have

n 2n 2 n +1
lim x = ∞ , lim x = ∞ lim x = −∞
x →∞ x →−∞ x →−∞

for all positive integers n . Using this and law 10 above, we get that for all positive integers m, n
n 2n 2 n +1
lim x m = ∞ , lim x 2 m +1 = ∞ lim x 2 m +1 = − ∞
x →∞ x →−∞ x →−∞

Example Evaluate

, , ,
Example Evaluate
 √ √
,, lim ( x2 + x −x2 − 2x)
x→∞
Note we can also use the squeeze theorem when calculating limits at ∞. Example Find

, , .
if they exist.
Limits of Polynomials at Infinity and minus infinity
Let
P(x) = a0 + a1x + a2x2 + ··· + anxn
be a polynomial function. Then the behavior of P(x) at ±∞ is the same as that of its highest term.
That is

and lim .

(To prove this consider the limit lim

Example Find lim x4 + 2x + 1, lim 2x3 + x2 + 1, lim −3x5 + 10x2 + 4562x + 1, lim (x − 2)3(x + 1)2(x −
1)5
x→∞ x→−∞ x→∞ x→∞

Note that we can use the following short cut for calculating limits of rational functions as x → ±∞:
axn + lin. comb. of lower powers axn lim = lim
m m
x→±∞ bx + lin. comb. of lower powers x→±∞ bx
where m and n are positive integers.

ASSIGNMENT: Think about it!

Compute each of the following limits:
1. lim 3x4
x→−∞
2. lim (-2x5)
x→−∞
3. lim 4x3
x→−∞
4. lim 23-1
x→−∞ 52+1
5. lim 3
x→−∞ x5

ASSESSMENT: You can do it!

Evaluate each of the following limits:
1. For f(x)=8x+9x3−11x5

a. limf(x)
x→∞
b. limf(x)
x→−∞

2. For h(t)=10t2+t4+6t−2

a. limf(x)
x→−∞
b. limf(x)
x→−∞

3. For g(z)=7+8z+3√z4

a. limf(x)
x→−∞
b. limf(x)
x→−∞

GROUP WORK: Work Together!

1. For f(x)=4x7−18x3+9 evaluate each of the following limits.
a. limf(x)
x→−∞
b. limf(x)
x→−∞
2. For h(t)=3√t+12t−2t2 evaluate each of the following limits.
a. limf(x)
x→−∞
b. limf(x)
x→−∞

READINGS
Find more related about infinity at limit at:

• http://tutorial.math.lamar.edu/ProblemsNS/CalcI/LimitsAtInfinityI.aspx
• http://www.math.drexel.edu/~mfm74/MFM/Teaching_files/Limits%20at%20Infinity.pdf
• https://web.auburn.edu/holmerr/1617/Textbook/limatinfty-screen.pdf
REFERENCES
Cdn.kutasoftware.com. (2019). Limit at infinity. [online] Available at:
https://cdn.kutasoftware.com/Worksheets/Calc/01%20-%20Limits%20at%20Infinity.pdf
[Accessed 23 Sep. 2019].
Cliffsnotes.com. (2016). Limits at Infinity. [online] Available at:
https://www.cliffsnotes.com/study-guides/calculus/calculus/limits/limits-at-infinity [Accessed
23 Sep. 2019].
Dawkins, P. (2007). Basic Calculus Book. [online] Notendur.hi.is. Available at:
https://notendur.hi.is/adl2/CalcI_Complete.pdf [Accessed 23 Sep. 2019].
Www3.nd.edu. (2019). Infinity at limit. [online] Available at:
https://www3.nd.edu/~apilking/Math10560/Calc1Lectures/19.%20Limits%20at%20infinity.p
df [Accessed 23 Sep. 2019].
ANSWER KEY
ASSESSMENT TO PRIOR KNOWLEDGE

1. INFINITY IS THE IDEA OF SOMETHING THAT HAS NO END

2. ∞
3. WE THEN SAY THAT THE VALUES OF F(X) BECOME INFINITE, OR TEND
TO INFINITY. WE SAY THAT AS X APPROACHES 0, THE LIMIT OF F(X) IS INFINITY.
NOW A LIMIT IS A NUMBER. SO WHEN WE SAY THAT THE LIMIT OF F(X)
IS INFINITY, WE MEAN THERE IS NO LIMIT TO ITS VALUES.

ASSIGMENT

1. −∞
2. ∞
3. −∞
4. −∞
5. −∞

ASSESSMENT

1. ∞
2. ∞
3. −∞

GROUP WORK

1. −∞
2. −∞