Science, Technology, Engineering and Mathematics

Basic Calculus
SY 2020 – 2021

Unit I: Limits and Continuity

Lesson 1.3: Infinite Limits

Introduction

In evaluating limits, the limit laws, rules on one-

sided limits and techniques in evaluating limits are
essential. But what if the limits of the given
function approaching a certain value of x cannot
apply these common rules and technique? What if
the value is an increasingly large number? A large
magnitude negative number? In this lesson we will
be learning about the infinite limits approaching a
certain value of x.

Learning Objectives

After studying this completing this module with 80% to 100% accuracy, you are expected
to:

a) illustrate the infinite limits; and

b) apply the infinite limit theorems in evaluating the limit of algebraic functions
(polynomial, rational and radical).

Pre-requisite Skills

To be successful in this module, you’ll need to master these skills and be able to apply in
problem solving situations.

Review: Large Numbers

It will be helpful to remember couple of simple things about fractions or ratios. Let’s use 0+ to
indicate a small positive number and 0- to indicate a small-magnitude negative number (large
negative number).
Examples:
positive number 14 14 14
a. = large positive number = 14, = 140, = 1, 400, ...
0+ 1 0.1 0.01

positive number 60 60 60
b. = large – magnitude = −60, = −600, = −6000,...
0− −1 −0.1 −0.01
negative number
negative number −45 −45 −45
= −45, = −450, = −4500,...
c. = large – magnitude
0+ 1 0.1 0.01
negative number
−843 −843 −843
negative number = 843, = 8430, = 84300,
d. = large positive number −1 −0.1 −0.01
0−
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Observe that (a) when a positive number is divided to a smaller positive number, the
resulting number is a larger positive number, on the other hand; (b) when it is divided to a smaller
magnitude negative number (larger negative number), the resulting number is a larger magnitude
negative number. (c) When a negative number is divided to a smaller positive number, the
resulting number is a larger magnitude negative number, in contrast, (d) when it is divided to a
larger magnitude negative number (smaller negative number), the resulting number is a larger
positive number. Now let’s look at an actual function.

Illustrative Example:
1
Let 𝑓(𝑥) = 𝑥−1. Examine the behavior of f near x = 1.

Solution: f (x) is a rational function and the point x = 1 is not in its domain. However, near x = 1,
we can make a table of values and plot its graph.

From the left of 1;

x<1 f(x)
0.9 -10
0.99 -100
0.999 -1000
0.9999 -10000
0.99999 -100000

From the right of 1;

x>1 f(x)
1.1 10
1.01 100
1.001 1000 1
as x → 1− , we see f ( x) → −
and as x → 1+ ,
1.0001 10000 0
1
1.00001 100000 we see f ( x) → .
0+

You may recognize the graph of f (x) above as having a vertical asymptote at x = 1. However,
when we are interested in the values of a function near some point, we should realize that we are
1 1
talking about limits. In the case of 𝑓(𝑥) = 𝑥−1, clearly lim 𝑥−1 does not exist. But we can still say
𝑥→1

something useful about the behavior of the function near 1. Because the values of f(x) grow
arbitrarily large or increase without bound as x→1+, we write lim+ 𝑓(𝑥) = +∞. The infinity
𝑥→1

symbol means that f(x) is getting large. It is not a number and the limit still does not exist in the
original sense of the term. Likewise, we write lim− 𝑓(𝑥) = −∞ because the values of f(x) decrease
𝑥→1

without bound as x→1 . -

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SY 2020 – 2021
Infinite Limits

The following definitions of various infinite limits are informal, but for most situations in this
subject, they will be adequate.

Suppose that f is defined for all x near a. We write lim 𝑓(𝑥) = +∞ and say that the limit
𝑥→𝑎

of f(x) as x approaches a is infinity if f(x) becomes arbitrarily large for all x sufficiently close
to (but not equal to) a.

We write lim 𝑓(𝑥) = −∞ and say that the limit of f(x) as x approaches a is negative
𝑥→𝑎

infinity if f(x) is negative and becomes arbitrarily large in magnitude for all x sufficiently close
to (but not equal to) a.

2
Determine lim |𝑥|.
𝑥→0

2
Solution: Notice that this limit does not exist in the usual sense since it is of the form 0. Intuitively,
2
we should recognize that the values of |𝑥| are becoming large in magnitude as x→0. We will deal

with most infinite limits either graphically or informally. A table of values and a quick plot
confirms this.

From the left of 1;

x<1 f(x)

-0.1 20

-0.01 200

-0.001 2000

-0.0001 20000

From the right of 1;

x>1 f(x)
1.1 10
1.01 100
1.001 1000
1.0001 10000 as x→0 we see f (x) increases without bound so

1.00001 100000 lim 𝑓(𝑥) = +∞.

𝑥→0

2
Thus, lim |𝑥| = +∞.
𝑥→0
 Science, Technology, Engineering and Mathematics
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SY 2020 – 2021
In the above example, f increased without bound from both sides of 0. In the first example, f
increased without bound from right sides of 1 and decreased without bound from left side.
Naturally, this leads to the notion of one-sided infinite limits. There are four possible behaviors
that can occur.

Suppose that f is defined for all x near a with x > a.

• We write lim+ 𝑓(𝑥) = +∞ if f (x) becomes arbitrarily large for all x sufficiently close to a
𝑥→𝑎
with x > a.
• Similarly, we write lim+ 𝑓(𝑥) = −∞ if f (x) is negative and becomes arbitrarily large in
𝑥→𝑎
magnitude for all x sufficiently close to a with x > a.
Now suppose that f is defined for x near a with x < a.

• We write lim− 𝑓(𝑥) = +∞ if f(x) becomes arbitrarily large for all x sufficiently close to a
𝑥→𝑎
with x < a.
• Similarly, we write lim− 𝑓(𝑥) = −∞ if f (x) is negative and becomes arbitrarily large in
𝑥→𝑎
magnitude for all x sufficiently close to a with x < a.

1 1
Note: In the first example, we saw that lim+ 𝑥−1 = +∞ and lim− 𝑥−1 = −∞. These limits still do
𝑥→1 𝑥→1
not exist in the usual sense. ±∞ is just a short-hand notation to describe the behavior of a function
near a point.

Tip: Most infinite limits should be evaluated as one-sided limits. Often, a function will exhibit a
very different behavior on either side of a point where the function is becoming unbounded.

Vertical Asymptotes

At a point where a function becomes large in magnitude, the graph appears almost vertical. To
describe this, we use the following terminology.

If lim+ 𝑓(𝑥) = ±∞ or if lim− 𝑓(𝑥) = ±∞, we say that the line x = a is a vertical asymptote
𝑥→𝑎 𝑥→𝑎

of f.

Many of the infinite limits that we will need to determine will arise from rational functions or other
functions that involve quotients. We look for infinite limits where such functions are not defined.
This typically means locating points where denominators of such functions are 0. Since graphing
may be time consuming (and since calculators sometimes give inaccurate graphs), we will use the
simple analytic method outlined in Section 0 a few pages back. Let’s try several examples.
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Situational Problems Determining Infinite Limits

Practice Problem #1
𝑥+1
Find the vertical asymptote and infinite limits for 𝑓(𝑥) = 𝑥−2.

Solution: This is a rational function which is continuous for all x except where the denominator
is 0, namely, x = 2. Examine the one-sided limit.

→3

x +1
lim− =−
x→ 2 x−2
→ 0−

The numerator approaches 3 while the denominator is negative and approaching 0. So, the quotient
is negative and large in magnitude. Similarly,

→3

x +1
lim+ =+
x→ 2 x−2
→ 0+

Either one-sided limit shows that x = 2 is a vertical asymptote.

𝑥+1
However, lim 𝑥−2 DNE since the left-hand limit is not equal to the right-hand limit.
𝑥→2

Practice Problem #2
6
Evaluate lim 𝑥 2 .
𝑥→0

Solution: As with previous example, let’s start off by looking the two one-sided limits. Once we
have those, we’ll be able to determine a value for the limit.

So, let’s look at the right-hand limit first and as noted above let’s see if we can figure out what
each limit will be doing without plugging in any values of x into the function. As we take smaller
and smaller values of x, while staying positive, squaring them will only make them smaller (recall
squaring a number between zero and one will make it smaller) and of course it will stay
 Science, Technology, Engineering and Mathematics
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SY 2020 – 2021
positive. So, we have a positive constant divided by an increasingly small positive number. The
result should then be an increasingly large positive number. It looks like we should have the
following value for the right-hand limit in this case,
6
lim+ = +∞
𝑥→0 𝑥2
Now, let’s take a look at the left-hand limit. In this case we’re going to take smaller and smaller
values of x, while staying negative this time. When we square them they’ll get smaller, but upon
squaring the result is now positive. So, we have a positive constant divided by an increasingly
small positive number. The result, as with the right-hand limit, will be an increasingly large
positive number and so the left-hand limit will be,
6
lim− = +∞
𝑥→0 𝑥2
Now, in this example, unlike the first one, the limit will exist and be infinity since the two one-
6
sided limits both exist and have the same value. Looking at the graph below, it shows that lim 𝑥 2 =
𝑥→0

+∞.

LIMIT THEOREM

If r is any positive integer, then

1 1  − if r is odd
lim+ = + and lim =
+ if r is even
r
x →0 xr x →0− x

Illustration 1:
Determine:
3 3 3
1. lim+ 2. lim− 3.lim
x →0 x2 x →0 x2 x →0 x2

3 3
1. lim− = + 2. lim+ = +
x →0 x2 x →0 x2

3 3 3
3.lim = + ( since lim− = lim+ 2 = + )
x →0 x2 x →0 x 2
x →0 x
 Science, Technology, Engineering and Mathematics
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LIMIT THEOREMS

If a is any real number and if lim f ( x) = 0 and lim g ( x) = c, where c is a constant not equal to 0, then
x →a x →a

g ( x)
❖ If c > 0 and if f ( x) → 0 through positive values of f (x), lim = +
x →a f ( x)
g ( x)
❖ If c > 0 and if f ( x) → 0 through negative of f (x), lim = −
x →a f ( x)
g ( x)
❖ If c < 0 and if f ( x) → 0 through positive values of f (x), lim = +
x →a f ( x)
g ( x)
❖ If c < 0 and if f ( x) → 0 through negative of f (x), lim = −
x →a f ( x)

Illustration 2:
3x
Evaluate lim− .
x→ 4 x−4
Solution:
lim 3x = 12 c0
x → 4−

lim ( x − 4 ) = 0 f ( x ) → 0 through negative values of f ( x )

x → 4−

2x
 lim+ =−
x→ 1 x −1
Illustration 3:
3x
Evaluate lim+ .
x→ 4 x−4
Solution:
lim 3x = 12 c0
x → 4−

lim ( x − 4 ) = 0 f ( x ) → 0 through positive values of f ( x )

x → 4−

2x
 lim+ =+
x→ 1 x −1
 Science, Technology, Engineering and Mathematics
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Key Concepts
✓ Infinite limits are limits of a function that approaches a certain value of x equal to
infinity.
✓ Most infinite limits should be evaluated as one-sided limits. Often, a function will exhibit
a very different behavior on either side of a point where the function is becoming
unbounded.
✓ Infinite limits are only being applied when substitution, factoring or rationalization are not
applicable since the resulting limits are still not undefined.
✓ If lim+ 𝑓(𝑥) = ±∞ or if lim− 𝑓(𝑥) = ±∞, we say that the line x = a is a vertical
𝑥→𝑎 𝑥→𝑎

asymptote of f.
1 1  − if r is odd
✓ If r is any positive integer, then lim+ = + and lim− r =  .
+ if r is even
r x →0 x
x →0 x

✓ If a is any real number and if lim f ( x) = 0 and lim g ( x) = c, where c is a constant not
x →a x →a

equal to 0, then
g ( x)
❖ If c > 0 and if f ( x) → 0 through positive values of f (x), lim = +
x →a f ( x)
g ( x)
❖ If c > 0 and if f ( x) → 0 through negative of f (x), lim = −
x →a f ( x)
g ( x)
❖ If c < 0 and if f ( x) → 0 through positive values of f (x), lim = +
x →a f ( x)
g ( x)
❖ If c < 0 and if f ( x) → 0 through negative of f (x), lim = −
x →a f ( x)

Learning Activity

Tutorial Videos

Here are some YouTube videos you can watch to help you to have a better understanding
about the lesson.

✓ Infinite Limits and Vertical Asymptotes

https://www.youtube.com/watch?v=xvFqomOpLrs (until 14: 42 mins)

References

YouTube Videos and Websites:

➢ Cliffnotes.com (2020, May 8). Infinite Limits. Cliffnotes.com.

https://www.cliffsnotes.com/study-guides/calculus/calculus/limits/infinite-limits
➢ Organic Chemistry (2017, June 5). Infinite Limits and Vertical Asymptotes. YouTube.
https://www.youtube.com/watch?v=xvFqomOpLrs
 Science, Technology, Engineering and Mathematics
Basic Calculus
SY 2020 – 2021
➢ Mathwarehouse.com (2020, May 8). Infinite Limits-Skills. Mathwarehouse.com.
https://www.mathwarehouse.com/calculus/limits/limits-infinite-skills.php

Got some questions and

clarifications? You may message
your subject teachers in Canvass
inbox or MS Teams. Or you can
post it on our discussion board.
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Enrichment Activity

A. Consider the graph of f(x) given below and compute the limits.

1. lim 𝑓(𝑥) 2. lim+ 𝑓(𝑥) 3. lim 𝑓(𝑥)

𝑥→−1− 𝑥→−1 𝑥→−1

4. lim 𝑓(𝑥) 5. lim+ 𝑓(𝑥) 6. lim 𝑓(𝑥)

𝑥→2− 𝑥→2 𝑥→2

B. Consider the graph of f(x) given below.

At what values of x does f (x) has an infinite limit (as x approaches this value)? Write down the
side limits.

C. Evaluate the given limits below. (1 pt. each)

127 x+5
1. lim+ 4. lim−
x →0 x34 x →2 x2 − 4

1028 x+5
2. lim− 5. lim+
x →0 x121 x →2 x2 − 4

12345 x+5
3. lim− 6. lim
x →0 x136 x →2 x2 − 4

Note: Please combine this Enrichment Activity 2.2 to Enrichment Activity 2.1 (on Module 1.2)
upon submission.
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NAME: _______________________________________ SECTION: ______________

BASIC CALCULUS
ENRICHMENT ACTIVITY #2.2
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