Infinity Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Infinity.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A concept representing a quantity that grows without bound — infinity is not a real number but a description of unbounded behavior.

Going on forever without end. Infinity is a direction or limiting idea, not a number you can reach or write down.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Infinity describes behavior that increases without bound; it's a limiting idea you approach, never a value you reach or do arithmetic on.

Common stuck point: The procedure for infinity is the easy part; the trap is writing ∞−∞=0\infty-\infty=0. Asking "Am I describing endless, unbounded growth or behavior at the edge of a domain, rather than computing with a real number?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I describing endless, unbounded growth or behavior at the edge of a domain, rather than computing with a real number?

Worked Examples

Example 1

easy
Evaluate lim⁡x→∞3x2+5x2−1\lim_{x \to \infty} \frac{3x^2 + 5}{x^2 - 1}.

Answer

33

First step

1
Divide numerator and denominator by the highest power of xx in the denominator, x2x^2.

Full solution

  1. 2
    Numerator: 3x2+5x2=3+5x2\frac{3x^2 + 5}{x^2} = 3 + \frac{5}{x^2}. Denominator: x2−1x2=1−1x2\frac{x^2-1}{x^2} = 1 - \frac{1}{x^2}.
  2. 3
    As x→∞x \to \infty, 5x2→0\frac{5}{x^2} \to 0 and 1x2→0\frac{1}{x^2} \to 0.
  3. 4
    Limit: 3+01−0=3\frac{3 + 0}{1 - 0} = 3.
For rational functions at infinity, divide by the highest power of xx in the denominator. Terms with xx in the denominator vanish, leaving only the ratio of leading coefficients. When degrees are equal, the limit is the ratio of leading coefficients.

Example 2

medium
Evaluate lim⁡x→∞2x3−x5x2+3\lim_{x \to \infty} \frac{2x^3 - x}{5x^2 + 3}.

Example 3

medium
Evaluate lim⁡x→∞(x2+3x−x)\lim_{x \to \infty}(\sqrt{x^2 + 3x} - x).

Example 4

medium
Why is 0⋅∞0 \cdot \infty an indeterminate form? Give two limits that show different results.

Example 5

hard
Evaluate lim⁡x→∞(x+2x−1)x\lim_{x \to \infty}\left(\frac{x+2}{x-1}\right)^x.

Example 6

challenge
Evaluate lim⁡n→∞nn\lim_{n \to \infty} \sqrt[n]{n}.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Evaluate lim⁡x→∞1x3\lim_{x \to \infty} \frac{1}{x^3}.

Example 2

medium
Evaluate lim⁡x→0+1x2\lim_{x \to 0^+} \frac{1}{x^2}.

Example 3

easy
True or false: ∞\infty is a real number you can add and subtract like any other.

Example 4

easy
As xx grows larger and larger without bound, what happens to f(x)=x2f(x)=x^2?

Example 5

easy
What is lim⁡x→0+1x\lim_{x\to0^+}\frac{1}{x} (as xx approaches 0 from the right)?

Example 6

easy
What is lim⁡x→0−1x\lim_{x\to0^-}\frac{1}{x} (from the left)?

Example 7

easy
Does lim⁡x→01x2\lim_{x\to0}\frac{1}{x^2} exist as a finite number?

Example 8

easy
Is the expression ∞∞\frac{\infty}{\infty} automatically equal to 1?

Example 9

easy
Counting numbers go 1, 2, 3, ... — is there a largest counting number?

Example 10

easy
As n→∞n\to\infty, what does the sequence an=1na_n=\frac{1}{n} approach?

Example 11

medium
Evaluate lim⁡x→∞3x2+5x2−2\lim_{x\to\infty}\frac{3x^2+5}{x^2-2}.

Example 12

medium
Evaluate lim⁡x→∞(x2+x−x)\lim_{x\to\infty}\big(\sqrt{x^2+x}-x\big).

Example 13

medium
The geometric series 12+14+18+⋯\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots has infinitely many terms. What is its sum?

Example 14

medium
Explain why 0⋅∞0\cdot\infty is called an indeterminate form. Give two products that show different results.

Example 15

medium
Evaluate lim⁡x→∞2x+1x2+3\lim_{x\to\infty}\frac{2x+1}{x^2+3}.

Example 16

medium
A ball is dropped and each bounce reaches half the previous height, starting at 4 m. What total vertical distance (down + up) does it travel?

Example 17

challenge
Show that lim⁡x→∞ln⁡xx=0\lim_{x\to\infty}\frac{\ln x}{x}=0, explaining what it says about growth rates.

Example 18

challenge
Determine lim⁡x→∞(1+1x)x\lim_{x\to\infty}\left(1+\frac{1}{x}\right)^x and explain why the answer is finite.

Example 19

challenge
Explain why ∞−∞\infty-\infty is indeterminate, and construct examples giving the values 5 and −∞-\infty.

Example 20

medium
Evaluate lim⁡x→∞5x3−x2x3+7\lim_{x\to\infty}\frac{5x^3-x}{2x^3+7}.

Example 21

medium
Evaluate lim⁡x→∞x2+1x+3\lim_{x\to\infty}\frac{x^2+1}{x+3}.

Example 22

medium
As n→∞n\to\infty, does an=(−1)nna_n=\frac{(-1)^n}{n} converge? To what?

Example 23

easy
Evaluate lim⁡x→∞4x\lim_{x \to \infty} \frac{4}{x}.

Example 24

easy
Evaluate lim⁡x→∞(7−2x)\lim_{x \to \infty}(7 - \tfrac{2}{x}).

Example 25

easy
Evaluate lim⁡x→0+1x4\lim_{x \to 0^+} \frac{1}{x^4}.

Example 26

easy
Evaluate lim⁡x→∞e−x\lim_{x \to \infty} e^{-x}.

Example 27

easy
Evaluate lim⁡x→∞(3+5x2)\lim_{x \to \infty}\left(3 + \frac{5}{x^2}\right).

Example 28

medium
Evaluate lim⁡x→∞4x3+2xx3−x2+1\lim_{x \to \infty} \frac{4x^3 + 2x}{x^3 - x^2 + 1}.

Example 29

medium
Evaluate lim⁡x→∞6x2+12x3−x\lim_{x \to \infty} \frac{6x^2 + 1}{2x^3 - x}.

Example 30

medium
Evaluate lim⁡x→1+1x−1\lim_{x \to 1^+} \frac{1}{x - 1}.

Example 31

medium
Evaluate lim⁡x→1−1x−1\lim_{x \to 1^-} \frac{1}{x - 1}.

Example 32

medium
Evaluate lim⁡n→∞(1−1n)n\lim_{n \to \infty}\left(1 - \frac{1}{n}\right)^n.

Example 33

medium
Find the horizontal asymptote of f(x)=3x2−1x2+5f(x) = \frac{3x^2 - 1}{x^2 + 5}.

Example 34

hard
Evaluate lim⁡x→∞ln⁡xx1/2\lim_{x \to \infty} \frac{\ln x}{x^{1/2}}.

Example 35

hard
Evaluate lim⁡x→∞xsin⁡(1/x)\lim_{x \to \infty} x \sin(1/x).

Example 36

hard
Find the horizontal asymptote of f(x)=4x2+1x+3f(x) = \frac{\sqrt{4x^2 + 1}}{x + 3} as x→+∞x \to +\infty.

Example 37

hard
Find lim⁡x→−∞4x2+1x+3\lim_{x \to -\infty} \frac{\sqrt{4x^2 + 1}}{x + 3}.

Example 38

challenge
Evaluate lim⁡x→0+xln⁡x\lim_{x \to 0^+} x \ln x.

Example 39

challenge
Evaluate lim⁡x→∞exx10\lim_{x \to \infty} \frac{e^x}{x^{10}}.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

limit