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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→∞f(x)=L\lim_{x \to \infty} f(x) = L and lim⁡x→−∞f(x)=L\lim_{x \to -\infty} f(x) = L mean
  • Use 1xn→0\frac{1}{x^n} \to 0 and e−x→0e^{-x} \to 0 to evaluate simple limits at infinity
  • Evaluate limits of rational functions by dividing by the highest power of xx
  • 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→∞x \to \infty)
  • • Limits by Direct Substitution
  • • Degree and leading coefficient of a polynomial
  • • Simplifying fractions and working with powers of xx
  • • Basic properties of the exponential function exe^x
Discussion Starters
  • 1. Why does 1x\frac{1}{x} get close to 00 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 xx?
Common Misconceptions

∞∞=1\frac{\infty}{\infty} = 1 and ∞−∞=0\infty - \infty = 0

Remediation: Show three fractions whose parts all grow without bound but have limits 00, 33 and none. Make a table at x=10,100,1000x = 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 2−8x3=−8x3+22 - 8x^3 = -8x^3 + 2.

A horizontal asymptote is a line the graph never touches

Remediation: Graph 3x+1x2+1\frac{3x + 1}{x^2 + 1} and point out where it crosses y=0y = 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=10,100,1000x = 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→∞9x2+1x+2\lim_{x \to \infty} \frac{\sqrt{9x^2 + 1}}{x + 2} and compare it with x→−∞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 aa in ax2+1x2+3\frac{ax^2 + 1}{x^2 + 3} and watch the horizontal asymptote y=ay = a follow it

  • activityDegree Race

    Students evaluate numerator and denominator at x=10,100,1000x = 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)f(x) when xx grows without bound (or decreases without bound).
lim⁡x→∞f(x)=L\lim_{x \to \infty} f(x) = L
means: as xx takes larger and larger values (10,100,1000,…10, 100, 1000, \ldots), the values f(x)f(x) get as close to LL as we like. In the same way, lim⁡x→−∞f(x)=L\lim_{x \to -\infty} f(x) = L looks at x=−10,−100,−1000,…x = -10, -100, -1000, \ldots
When such a finite limit exists, the line y=Ly = L is a horizontal asymptote of the graph.
The basic building block: for any positive power nn,
lim⁡x→±∞1xn=0\lim_{x \to \pm\infty} \frac{1}{x^n} = 0
A fixed number divided by a huge number is tiny. Also lim⁡x→∞e−x=0\lim_{x \to \infty} e^{-x} = 0, because e−x=1exe^{-x} = \frac{1}{e^x}.
Rational functions: compare the degree of the numerator (nn) with the degree of the denominator (mm):
  • n<mn < m: the limit is 00 (asymptote y=0y = 0)
  • n=mn = m: the limit is the ratio of the leading coefficients
  • n>mn > m: there is no finite limit (no horizontal asymptote)
Why it works: divide every term by the highest power of xx in the denominator. Every term of the form cxk\frac{c}{x^k} goes to 00 and only the leading terms are left. For example, 2x2+1x2+3=2+1x21+3x2→21=2\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→∞(3+5x2)\lim_{x \to \infty} \left(3 + \frac{5}{x^2}\right).
1

Try a few large values of x

x=10x = 10: 3+5100=3.053 + \frac{5}{100} = 3.05; x=100x = 100: 3+510000=3.00053 + \frac{5}{10000} = 3.0005 → The values approach 33
2

Use the basic rule

5x2=5⋅1x2\frac{5}{x^2} = 5 \cdot \frac{1}{x^2} and 1x2→0\frac{1}{x^2} \to 0 → 5x2→0\frac{5}{x^2} \to 0
3

Add the limits

3+03 + 0 → 33

Common Mistakes

Plugging in infinity: ∞∞=1\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 xx first. For 6x2−12x2+5\frac{6x^2 - 1}{2x^2 + 5} the limit is 33, not 11.

Using the first written coefficient instead of the leading one: 4x3−x2−8x3→42=2\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 2−8x32 - 8x^3 is −8x3-8x^3, so the limit is 4−8=−12\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 00.

Correct: Smaller degree on top: limit 00. 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=3x+1x2+1y = \frac{3x + 1}{x^2 + 1} has the asymptote y=0y = 0 and still crosses it at x=−13x = -\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→∞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−0.05tT(t) = 22 + 60e^{-0.05t} (degrees Celsius, tt in minutes). Since e−0.05t→0e^{-0.05t} \to 0, the tea approaches 22°C22°C.

1Try It Yourself

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

What temperature does the coffee approach in the long run?

Step 1: Write the mathematical expression

Find lim⁡t→∞T(t)\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 300+2xx\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 xx mugs is C(x)=5000+12xxC(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→∞C(x)\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 50t50t grams while the volume is 100+2t100 + 2t litres, the concentration 50t100+2t\frac{50t}{100 + 2t} approaches 502=25\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 tt minutes the concentration is C(t)=120t200+4tC(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→∞C(t)\lim_{t \to \infty} C(t) in grams per litre:

Key Takeaways

  • 1lim⁡x→∞f(x)=L\lim_{x \to \infty} f(x) = L means f(x)f(x) gets as close to LL as we like when xx grows without bound; x→−∞x \to -\infty works the same way to the left
  • 2Basic facts: 1xn→0\frac{1}{x^n} \to 0 for any positive nn as x→±∞x \to \pm\infty, and e−x→0e^{-x} \to 0 as x→∞x \to \infty
  • 3To find the limit of a rational function, divide every term by the highest power of xx in the denominator
  • 4Degree on top smaller: limit 00. Equal degrees: ratio of leading coefficients. Degree on top bigger: no finite limit
  • 5A finite limit LL at infinity means y=Ly = 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=1000x = 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 33 instead of 2.99922.9992.

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→0e^{-x} \to 0 as x→∞x \to \infty, but e−xe^{-x} grows without bound as x→−∞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 3x+1x2+1\frac{3x + 1}{x^2 + 1} crosses y=0y = 0 at x=−13x = -\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 x3x2+1\frac{x^3}{x^2 + 1} that line is y=xy = x. You find it with polynomial division.

Glossary

Limit at infinity
The value LL that f(x)f(x) approaches as x→∞x \to \infty or x→−∞x \to -\infty, written lim⁡x→∞f(x)=L\lim_{x \to \infty} f(x) = L
Horizontal asymptote
A horizontal line y=Ly = L that the graph approaches far to the left or right, where LL is a limit at infinity
Degree
The highest power of xx in a polynomial, for example 33 in 4x3−x4x^3 - x
Leading term
The term with the highest power of xx; its number is the leading coefficient (in 2−8x32 - 8x^3 the leading term is −8x3-8x^3)
End behavior
How the values of a function behave as xx grows without bound in either direction

Formula Card

Reciprocal powers

lim⁡x→±∞cxn=0(n>0)\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→∞e−x=0\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→±∞anxn+⋯bmxm+⋯=0n < 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→±∞anxn+⋯bmxm+⋯=anbmn = 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: no finite limitn > m: \ \text{no finite limit}

The values grow without bound in size, no horizontal asymptote

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