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.

Advanced30 min18 problems
Practice Problems

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.

Try it now

Find lim⁡x→∞7x3\lim_{x \to \infty} \frac{7}{x^3}.

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}.

Interactive Visual

Function graph

-40-30-20-10010203040-4-2024xyy = 2

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y = (a·x² + 1)/(x² + 3)Asymptote

Table of values

x-1000-100-100101001000
y21.99951.951460.3333331.951461.99952

Function graph

-20-15-10-505101520-505xyy = 0y = 3y = 2

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Asymptote

Table of values

x101001000
degree 1 / degree 20.3069310.0300970.003001
degree 2 / degree 22.921952.99922.99999
degree 3 / degree 29.9009999.99999.999
2 + 5e^(−x)2.0002322

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Interactive Grapher

-30-20-100102030-505xy

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y = 2x + 1

m=2, b=1

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Practice Problems

18 problems
Problem 1 of 18
Easy
Find lim⁡x→∞7x3\lim_{x \to \infty} \frac{7}{x^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

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.
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.
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.
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.
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

Today's challengeOne problem, three tries

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