Teacher Guide: Limits at Infinity

Find what a function approaches as x grows without bound, compare degrees of rational functions and connect limits at infinity to horizontal asymptotes.

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Learning Objectives

Explain what $\lim_{x \to \infty} f(x) = L$ and $\lim_{x \to -\infty} f(x) = L$ mean
Use $\frac{1}{x^n} \to 0$ and $e^{-x} \to 0$ to evaluate simple limits at infinity
Evaluate limits of rational functions by dividing by the highest power of $x$
Predict the limit of a rational function by comparing the degrees of numerator and denominator
Connect a finite limit at infinity to a horizontal asymptote of the graph

Prerequisites

• Introduction to Limits and Limit Notation (reading $x \to \infty$ )
• Limits by Direct Substitution
• Degree and leading coefficient of a polynomial
• Simplifying fractions and working with powers of $x$
• Basic properties of the exponential function $e^x$

Discussion Starters

1. Why does $\frac{1}{x}$ get close to $0$ but never reach it? What does that look like on the graph?
2. If the numerator and the denominator both grow without bound, how can the fraction still approach a finite number?
3. Can you think of a quantity in everyday life that settles at a fixed value in the long run?
4. Why is it fine for a graph to cross a horizontal asymptote but not a vertical one?
5. How would you explain to a friend why only the leading terms matter for very large $x$ ?

Common Misconceptions

$\frac{\infty}{\infty} = 1$ and $\infty - \infty = 0$

Remediation: Show three fractions whose parts all grow without bound but have limits $0$ , $3$ and none. Make a table at $x = 10, 100, 1000$ so students see different growth speeds.

The leading coefficient is the first number written

Remediation: Have students rewrite each polynomial in standard form (highest power first) before comparing, for example $2 - 8x^3 = -8x^3 + 2$ .

A horizontal asymptote is a line the graph never touches

Remediation: Graph $\frac{3x + 1}{x^2 + 1}$ and point out where it crosses $y = 0$ . Stress that the asymptote is about the ends of the graph only.

Differentiation Ideas

For Struggling Students:

• Start with tables of values at $x = 10, 100, 1000$ before any algebra
• Use only functions with equal degrees and positive coefficients at first
• Give a three-row card for the degree rules to keep beside the work

For On-Level Students:

• Practice dividing by the highest power for all three degree cases
• Include limits at $-\infty$ and negative leading coefficients
• Match rational functions to graphs using their horizontal asymptotes

For Advanced Students:

• Find the oblique asymptote when the numerator's degree is one higher, using polynomial division
• Investigate $\lim_{x \to \infty} \frac{\sqrt{9x^2 + 1}}{x + 2}$ and compare it with $x \to -\infty$
• Design a rational function with a given horizontal asymptote and a given zero

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
HSF-IF.B.4 (CCSS.MATH.CONTENT.HSF.IF.B.4)
Interpret key features of graphs and tables of a function, including end behavior
AP Calculus LIM-2.D (AP.CALC.LIM-2.D)
Interpret the behavior of functions using limits involving infinity

Lesson Resources

visualAsymptote Slider
Move the leading coefficient $a$ in $\frac{ax^2 + 1}{x^2 + 3}$ and watch the horizontal asymptote $y = a$ follow it
activityDegree Race
Students evaluate numerator and denominator at $x = 10, 100, 1000$ to see which part grows faster
worksheetThree Cases Sort
Sort 12 rational functions into limit 0, ratio of leading coefficients, or no finite limit

Lesson Content

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

Definition

A limit at infinity describes what happens to

f(x)

when

x

grows without bound (or decreases without bound).

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

means: as

x

takes larger and larger values (

10, 100, 1000, \ldots

), the values

f(x)

get as close to

L

as we like. In the same way,

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

looks at

x = -10, -100, -1000, \ldots

When such a finite limit exists, the line

y = L

is a horizontal asymptote of the graph.

The basic building block: for any positive power

n

\lim_{x \to \pm\infty} \frac{1}{x^n} = 0

A fixed number divided by a huge number is tiny. Also

\lim_{x \to \infty} e^{-x} = 0

, because

e^{-x} = \frac{1}{e^x}

Rational functions: compare the degree of the numerator (

n

) with the degree of the denominator (

m

$n < m$ : the limit is $0$ (asymptote $y = 0$ )
$n = m$ : the limit is the ratio of the leading coefficients
$n > m$ : there is no finite limit (no horizontal asymptote)

Why it works: divide every term by the highest power of

x

in the denominator. Every term of the form

\frac{c}{x^k}

goes to

0

and only the leading terms are left. For example,

\frac{2x^2 + 1}{x^2 + 3} = \frac{2 + \frac{1}{x^2}}{1 + \frac{3}{x^2}} \to \frac{2}{1} = 2

Worked Examples

Find $\lim_{x \to \infty} \left(3 + \frac{5}{x^2}\right)$ .

Try a few large values of x

$x = 10$ : $3 + \frac{5}{100} = 3.05$ ; $x = 100$ : $3 + \frac{5}{10000} = 3.0005$ → The values approach $3$

Use the basic rule

$\frac{5}{x^2} = 5 \cdot \frac{1}{x^2}$ and $\frac{1}{x^2} \to 0$ → $\frac{5}{x^2} \to 0$

Add the limits

$3 + 0$ → $3$

Answer: $\lim_{x \to \infty} \left(3 + \frac{5}{x^2}\right) = 3$ , so $y = 3$ is a horizontal asymptote.

Common Mistakes

Plugging in infinity: $\frac{\infty}{\infty} = 1$

Why it's wrong: $\infty$ is not a number, so you cannot substitute it or cancel it. Numerator and denominator can grow at very different speeds.

Correct: Divide by the highest power of $x$ first. For $\frac{6x^2 - 1}{2x^2 + 5}$ the limit is $3$ , not $1$ .

Using the first written coefficient instead of the leading one: $\frac{4x^3 - x}{2 - 8x^3} \to \frac{4}{2} = 2$

Why it's wrong: The leading coefficient belongs to the term with the highest power, wherever it stands in the expression.

Correct: The leading term of $2 - 8x^3$ is $-8x^3$ , so the limit is $\frac{4}{-8} = -\frac{1}{2}$ .

Swapping the rules: "top degree smaller means no limit"

Why it's wrong: When the denominator has the bigger degree, it grows much faster, which pushes the fraction toward $0$ .

Correct: Smaller degree on top: limit $0$ . Bigger degree on top: no finite limit. Equal degrees: ratio of leading coefficients.

Believing a graph can never cross its horizontal asymptote

Why it's wrong: A horizontal asymptote only describes the far left and far right of the graph. In the middle the curve may cross it.

Correct: $y = \frac{3x + 1}{x^2 + 1}$ has the asymptote $y = 0$ and still crosses it at $x = -\frac{1}{3}$ .

Why It Matters

Limits at infinity answer the question "what happens in the long run?":

Science: hot coffee cools toward room temperature; the limit of its temperature model is that room temperature
Business: the average cost per item falls toward the cost of materials as production grows, because fixed costs are spread thin
Medicine and chemistry: concentrations in a mixing tank or in the blood settle at a steady level
Graphing: horizontal asymptotes show the end behavior of rational functions at a glance
Calculus: improper integrals, series and many models in physics are built on limits at infinity

This lesson builds on Introduction to Limits and Limit Notation: there you met the symbol

x \to \infty

, here you learn to compute such limits quickly and reliably.

Real World Applications

Cooling Coffee

Newton's law of cooling says a hot drink cools toward the temperature of the room. The long-run temperature is a limit at infinity.

Example:

A cup of tea follows $T(t) = 22 + 60e^{-0.05t}$ (degrees Celsius, $t$ in minutes). Since $e^{-0.05t} \to 0$ , the tea approaches $22°C$ .

1Try It Yourself

Freshly made coffee has the temperature $T(t) = 20 + 70e^{-0.1t}$ degrees Celsius after $t$ minutes.

What temperature does the coffee approach in the long run?

Step 1: Write the mathematical expression

Find $\lim_{t \to \infty} T(t)$ :

Average Cost in a Workshop

Fixed costs (rent, machines) are shared by all items produced. The more items, the smaller each one's share, so the average cost approaches the cost per item.

Example:

A bakery with 300 dollars of daily fixed costs and 2 dollars of ingredients per cake has average cost $\frac{300 + 2x}{x}$ , which approaches 2 dollars per cake.

2Try It Yourself

A pottery workshop has fixed costs of 5000 dollars, and each mug costs 12 dollars in clay and glaze. The average cost of $x$ mugs is $C(x) = \frac{5000 + 12x}{x}$ dollars.

What does the average cost per mug approach as the number of mugs grows without bound?

Step 1: Write the mathematical expression

Find $\lim_{x \to \infty} C(x)$ in dollars:

Salt Concentration in a Tank

Chemical engineers model how a mixture changes over time. When salty water keeps flowing into a tank, the concentration settles at a steady value.

Example:

If salt in a tank grows like $50t$ grams while the volume is $100 + 2t$ litres, the concentration $\frac{50t}{100 + 2t}$ approaches $\frac{50}{2} = 25$ grams per litre.

3Try It Yourself

A tank holds 200 litres of pure water. Salt water with 30 grams of salt per litre flows in at 4 litres per minute (and nothing flows out). After $t$ minutes the concentration is $C(t) = \frac{120t}{200 + 4t}$ grams per litre.

What concentration does the tank approach in the long run?

Step 1: Write the mathematical expression

Find $\lim_{t \to \infty} C(t)$ in grams per litre:

Key Takeaways

1 $\lim_{x \to \infty} f(x) = L$ means $f(x)$ gets as close to $L$ as we like when $x$ grows without bound; $x \to -\infty$ works the same way to the left
2Basic facts: $\frac{1}{x^n} \to 0$ for any positive $n$ as $x \to \pm\infty$ , and $e^{-x} \to 0$ as $x \to \infty$
3To find the limit of a rational function, divide every term by the highest power of $x$ in the denominator
4Degree on top smaller: limit $0$ . Equal degrees: ratio of leading coefficients. Degree on top bigger: no finite limit
5A finite limit $L$ at infinity means $y = L$ is a horizontal asymptote; a graph may still cross it in the middle
6Never substitute $\infty$ : expressions like $\frac{\infty}{\infty}$ have no fixed value

Frequently Asked Questions

Can I just plug in a very big number instead of computing the limit?

A big number such as

x = 1000

is a good check and often shows the trend, but it is not a proof. The algebra (dividing by the highest power) tells you the exact value, for example exactly

3

instead of

2.9992

Are the limits at $\infty$ and at $-\infty$ always the same?

For rational functions with a finite limit, yes: both ends give the same horizontal asymptote. Other functions can differ. For example,

e^{-x} \to 0

x \to \infty

, but

e^{-x}

grows without bound as

x \to -\infty

Can a graph cross its horizontal asymptote?

Yes. The asymptote describes only the end behavior, far to the left or right. The graph of

\frac{3x + 1}{x^2 + 1}

crosses

y = 0

x = -\frac{1}{3}

and then approaches it again.

What happens when the numerator's degree is exactly one more than the denominator's?

This is an extension beyond this lesson. There is no horizontal asymptote, but the graph approaches a slanted line (an oblique asymptote). For

\frac{x^3}{x^2 + 1}

that line is

y = x

. You find it with polynomial division.

Glossary

Limit at infinity: The value $L$ that $f(x)$ approaches as $x \to \infty$ or $x \to -\infty$ , written $\lim_{x \to \infty} f(x) = L$
Horizontal asymptote: A horizontal line $y = L$ that the graph approaches far to the left or right, where $L$ is a limit at infinity
Degree: The highest power of $x$ in a polynomial, for example $3$ in $4x^3 - x$
Leading term: The term with the highest power of $x$ ; its number is the leading coefficient (in $2 - 8x^3$ the leading term is $-8x^3$ )
End behavior: How the values of a function behave as $x$ grows without bound in either direction

Formula Card

Reciprocal powers

\lim_{x \to \pm\infty} \frac{c}{x^n} = 0 \quad (n > 0)

A fixed number divided by an ever larger power shrinks to zero

Exponential decay

\lim_{x \to \infty} e^{-x} = 0

Since $e^{-x} = \frac{1}{e^x}$ and $e^x$ grows without bound

Smaller degree on top

n < m: \ \lim_{x \to \pm\infty} \frac{a_n x^n + \cdots}{b_m x^m + \cdots} = 0

Horizontal asymptote $y = 0$

Equal degrees

n = m: \ \lim_{x \to \pm\infty} \frac{a_n x^n + \cdots}{b_m x^m + \cdots} = \frac{a_n}{b_m}

Horizontal asymptote at the ratio of the leading coefficients

Bigger degree on top

n > m: \ \text{no finite limit}

The values grow without bound in size, no horizontal asymptote

Teacher Guide: Limits at Infinity

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

Cooling Coffee

Average Cost in a Workshop

Salt Concentration in a Tank

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Reciprocal powers

Exponential decay

Smaller degree on top

Equal degrees

Bigger degree on top

More in This Topic

Introduction to Limits

Limit Notation

Evaluating Limits Graphically

Evaluating Limits from Tables