Teacher Guide: Infinite Limits

Learn what it means when a function grows without bound near a point, how to find the sign with denominator analysis, and how infinite limits create vertical asymptotes.

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For Teachers

Learning Objectives

Interpret the notation $\lim_{x \to a} f(x) = \infty$ and $-\infty$ for one-sided and two-sided limits
Use sign analysis of the numerator and denominator to decide whether a limit is $\infty$ or $-\infty$
Decide whether a two-sided infinite limit can be written as $\pm\infty$ or does not exist
Find vertical asymptotes of rational functions and tell them apart from holes
Explain why a limit equal to $\infty$ is not a limit that exists as a number

Prerequisites

• The lessons "Limit Notation" and "One-Sided Limits"
• The lessons "Evaluating Limits from Tables" and "Limits by Direct Substitution"
• Factoring quadratic expressions
• Sign rules for multiplying and dividing negative numbers

Discussion Starters

1. Is "the limit equals infinity" the same as "the limit exists"? Why or why not?
2. Why can't we write $\lim_{x \to 0} \frac{1}{x} = \infty$ , while $\lim_{x \to 0} \frac{1}{x^2} = \infty$ is correct?
3. Both $\frac{x-2}{x-2}$ and $\frac{1}{x-2}$ have a zero denominator at $x = 2$ . Why does only one of them have a vertical asymptote?
4. Can you think of a quantity in real life that grows without bound as you get close to some value?

Common Misconceptions

An infinite limit means the limit exists and is a very big number

Remediation: Show a table for $\frac{1}{(x-2)^2}$ : whatever bound a student names (one million, one billion), find an $x$ close to $2$ that beats it. Conclude that no number works, so the limit does not exist as a number.

Every zero of the denominator gives a vertical asymptote

Remediation: Graph $\frac{x^2-4}{x-2}$ next to $\frac{1}{x-2}$ . The first is the line $y = x + 2$ with a hole at $x = 2$ ; only the second blows up. Make checking the numerator a fixed step.

If one side goes to $\infty$ , the two-sided limit is $\infty$

Remediation: Use the graph of $\frac{1}{x-2}$ and ask students to trace each branch toward $x = 2$ . Insist on writing both one-sided limits before writing a two-sided one.

Differentiation Ideas

For Struggling Students:

• Start with tables of values (1.9, 1.99, 2.01, 2.1) before any sign reasoning
• Give a sign chart template with boxes for numerator, denominator and quotient
• Use only denominators of the form $x - a$ and $(x - a)^2$ at first

For On-Level Students:

• Find vertical asymptotes and one-sided limits of rational functions with factorable denominators
• Classify each zero of the denominator as an asymptote or a hole
• Sketch the graph near each asymptote from the one-sided limits

For Advanced Students:

• Explore infinite limits of non-rational functions such as $\ln x$ as $x \to 0^+$ and $\tan x$ near $\frac{\pi}{2}$
• Prove $\lim_{x \to 0} \frac{1}{x^2} = \infty$ from the definition: for every $M > 0$ find a $\delta > 0$
• Build a rational function with given asymptotes, holes and one-sided behavior

Standards Alignment

HSF-IF.C.7d (CCSS.MATH.CONTENT.HSF.IF.C.7.D)
Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior
AP Calculus LIM-2.D (AP.CALC.LIM-2.D)
Interpret the behavior of functions using limits involving infinity

Lesson Resources

visualGraph of 1/(x - 2) and 1/(x - 2)²
Switch between the two graphs and compare the value tables on both sides of the asymptote $x = 2$
activitySign Chart Relay
Teams fill in the sign of numerator, denominator and quotient on each side of every zero of the denominator, then sketch the graph
worksheetAsymptote or Hole?
Ten rational functions: classify every zero of the denominator and give the one-sided limits

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

An infinite limit describes a function whose values grow without bound as

x

approaches a number

a

\lim_{x \to a} f(x) = \infty

means: as

x

gets close to

a

(from both sides),

f(x)

becomes larger than any number you can name. In the same way,

\lim_{x \to a} f(x) = -\infty

means

f(x)

becomes more negative than any number you can name.

One-sided versions work the same way:

\lim_{x \to a^-} f(x) = \infty

looks only at

x < a

, and

\lim_{x \to a^+} f(x) = -\infty

looks only at

x > a

Important:

\infty

is not a number. Writing

\lim_{x \to a} f(x) = \infty

tells you *how* the limit fails to exist as a real number: the values blow up in the positive direction. The limit does not exist as a number.

Vertical asymptote: the line

x = a

is a vertical asymptote of

f

if at least one of the one-sided limits at

a

\infty

-\infty

. For a rational function

\frac{p(x)}{q(x)}

this happens where

q(a) = 0

but

p(a) \neq 0

Sign analysis: when the numerator approaches a nonzero number and the denominator approaches

0

, the quotient grows without bound. Its sign comes from the signs of the two parts:

positive divided by a small positive number: $\infty$
positive divided by a small negative number: $-\infty$
negative divided by a small positive number: $-\infty$
negative divided by a small negative number: $\infty$

Worked Examples

Find $\lim_{x \to 2} \frac{1}{(x-2)^2}$ .

Try direct substitution

$\frac{1}{(2-2)^2} = \frac{1}{0}$ : the numerator is $1$ , the denominator is $0$ → Not a number: look closer

Make a table from the left

$x = 1.9$ : $\frac{1}{0.01} = 100$ ; $x = 1.99$ : $\frac{1}{0.0001} = 10000$ → Values grow without bound

Make a table from the right

$x = 2.1$ : $100$ ; $x = 2.01$ : $10000$ → Values grow without bound

Use the sign

$(x-2)^2$ is a square, so it is positive on both sides of $2$ → Both one-sided limits are $\infty$

Answer: $\lim_{x \to 2} \frac{1}{(x-2)^2} = \infty$ . Both sides blow up in the same direction, so we write $\infty$ for the two-sided limit, and $x = 2$ is a vertical asymptote.

Common Mistakes

Saying "the limit exists and equals $\infty$ "

Why it's wrong: $\infty$ is not a real number. Writing $= \infty$ only describes the way the function grows; the limit does not exist as a number.

Correct: Say " $f(x)$ grows without bound as $x \to a$ " or "the limit is $\infty$ ", and remember that a limit equal to $\infty$ is not a limit that exists.

Writing $\lim_{x \to 2} \frac{1}{x-2} = \infty$

Why it's wrong: From the left the values go to $-\infty$ , from the right to $\infty$ . A two-sided limit can only be written as $\infty$ when both sides go to $\infty$ .

Correct: Give both one-sided limits ( $-\infty$ and $\infty$ ) and say that the two-sided limit does not exist.

Calling every zero of the denominator a vertical asymptote

Why it's wrong: If the numerator is also $0$ there, the expression has the form $\frac{0}{0}$ and may simplify. Then the limit can be a number and the graph only has a hole.

Correct: Check the numerator first: denominator $0$ and numerator not $0$ means a vertical asymptote; both $0$ means simplify and look again.

Choosing the sign from the denominator only

Why it's wrong: A negative numerator flips the result: $\frac{-5}{\text{small positive}}$ goes to $-\infty$ , not $\infty$ .

Correct: Find the sign of the numerator and the sign of the denominator on the side you are looking at, then combine them like in division.

Why It Matters

Infinite limits describe situations where a quantity explodes as you approach a critical value:

Physics: the force between two charges, $F = \frac{k}{r^2}$ , grows without bound as the distance $r$ approaches $0$
Optics: when an object is placed close to the focal point of a lens, the image moves infinitely far away
Environmental science: the cost or time needed to remove the last few percent of a pollutant often grows without bound as you approach 100%
Graphing: infinite limits tell you exactly where the vertical asymptotes of a rational function are and which way the graph goes on each side

In calculus you will use infinite limits to sketch graphs, to describe discontinuities and to work with improper integrals.

Real World Applications

Electric Force at Short Distances

The force between two charged particles is inversely proportional to the square of their distance, so it grows without bound as the distance shrinks to zero.

Example:

With $F(r) = \frac{100}{r^2}$ (in newtons, $r$ in meters), halving the distance makes the force four times larger, and $\lim_{r \to 0^+} F(r) = \infty$ .

1Try It Yourself

Two charged particles push each other apart with a force of $F(r) = \frac{100}{r^2}$ newtons, where $r$ is the distance in meters.

How large is the force, in newtons, at a distance of $r = 0.1$ meters?

Step 1: Write the mathematical expression

Compute $F(0.1)$ :

Removing the Last Impurities

Water treatment gets harder and harder as you approach complete purity. Models of the effort often have a vertical asymptote at 100%.

Example:

If filtering $p$ percent of the impurities takes $T(p) = \frac{50p}{100-p}$ minutes, then $T(50) = 50$ minutes, but $T(90) = 450$ minutes.

2Try It Yourself

A filter needs $T(p) = \frac{50p}{100-p}$ minutes to remove $p$ percent of the impurities from a water tank.

What does $T(p)$ approach as $p \to 100^-$ ?

Step 1: Choose the right answer

Choose the value of $\lim_{p \to 100^-} T(p)$ :

Images Formed by a Lens

For a lens with focal length $f$, the image distance is $d_i = \frac{f \, d_o}{d_o - f}$. As the object moves toward the focal point, the image moves infinitely far away.

Example:

For $f = 10$ cm: an object at $d_o = 20$ cm gives an image at $d_i = \frac{200}{10} = 20$ cm, and $\lim_{d_o \to 10^+} d_i = \infty$ .

3Try It Yourself

A magnifying lens has focal length $10$ cm, so the image distance is $d_i = \frac{10 \, d_o}{d_o - 10}$ cm.

What is the image distance, in cm, when the object is at $d_o = 11$ cm?

Step 1: Write the mathematical expression

Compute $d_i$ for $d_o = 11$ :

Key Takeaways

1 $\lim_{x \to a} f(x) = \infty$ means $f(x)$ grows without bound as $x$ approaches $a$ ; $-\infty$ means it decreases without bound
2 $\infty$ is not a number: a limit equal to $\infty$ or $-\infty$ does not exist as a real number
3If the numerator approaches a nonzero number and the denominator approaches $0$ , the limit is infinite; sign analysis decides between $\infty$ and $-\infty$
4Check each side separately: if the left and right sides go in different directions, the two-sided limit does not exist and is not written as $\infty$
5 $x = a$ is a vertical asymptote when at least one one-sided limit at $a$ is $\infty$ or $-\infty$
6A zero of the denominator where the numerator is also $0$ may be a hole, not an asymptote: simplify first

Frequently Asked Questions

If the limit is $\infty$ , does the limit exist?

No. A limit exists only when it is a real number. Writing

\lim_{x \to a} f(x) = \infty

is a more precise way of saying "the limit does not exist because the values grow without bound in the positive direction".

How is an infinite limit different from a limit at infinity?

In an infinite limit,

x

approaches a number and

f(x)

blows up:

\lim_{x \to 2} \frac{1}{(x-2)^2} = \infty

. In a limit at infinity,

x

itself grows without bound:

\lim_{x \to \infty} \frac{1}{x} = 0

. The first gives vertical asymptotes, the second horizontal ones.

Can a graph cross a vertical asymptote?

No. At a vertical asymptote

x = a

the function values blow up near

a

, and

f(a)

itself is usually not defined. (Horizontal asymptotes are different: a graph can cross those.)

Why does an even power in the denominator give the same sign on both sides?

A square (or any even power) is never negative. Near

a

(x-a)^2

is a small positive number on both sides, so the sign of the quotient depends only on the numerator.

Glossary

Infinite limit: A limit where $f(x)$ grows (or decreases) without bound as $x \to a$ , written $\lim_{x \to a} f(x) = \infty$ or $-\infty$
Vertical asymptote: A vertical line $x = a$ that the graph approaches while its values go to $\infty$ or $-\infty$ on at least one side
Sign analysis: Finding whether the numerator and the denominator are positive or negative near $a$ to decide between $\infty$ and $-\infty$
Grows without bound: Becomes larger than any number you can name; this is what $\to \infty$ means
Does not exist (DNE): The limit is not a real number, for example because the values blow up or the two sides disagree

Formula Card

Infinite limit test

\lim_{x \to a^{\pm}} \frac{p(x)}{q(x)} = \pm\infty \text{ if } p(a) \neq 0,\ q(a) = 0

Nonzero numerator over a denominator approaching $0$: each one-sided limit is $\infty$ or $-\infty$, the signs decide which; the two-sided limit is $\pm\infty$ only when both sides agree

Odd power

\lim_{x \to a^-} \frac{1}{x-a} = -\infty, \quad \lim_{x \to a^+} \frac{1}{x-a} = \infty

The two sides go in opposite directions, so the two-sided limit does not exist

Even power

\lim_{x \to a} \frac{1}{(x-a)^2} = \infty

A square is positive on both sides, so both sides go to $\infty$

Vertical asymptote

x = a \text{ if } \lim_{x \to a^-} f(x) \text{ or } \lim_{x \to a^+} f(x) \text{ is } \pm\infty

One infinite one-sided limit is enough

Teacher Guide: Infinite Limits

Use this lesson with your class

For Teachers

For Struggling Students:

For On-Level Students:

For Advanced Students:

Lesson Content

Definition

Worked Examples

Common Mistakes

Why It Matters

Real World Applications

Electric Force at Short Distances

Removing the Last Impurities

Images Formed by a Lens

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Infinite limit test

Odd power

Even power

Vertical asymptote

More in This Topic

Introduction to Limits

Limit Notation

Evaluating Limits Graphically

Evaluating Limits from Tables