Infinity Examples: 45 Problems with Answers

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 $\infty-\infty=0$ . 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 \to \infty} \frac{3 x^{2} + 5}{x^{2} - 1}

Answer

3

First step

Divide numerator and denominator by the highest power of

x

in the denominator,

x^2

Full solution

2
Numerator: $\frac{3x^2 + 5}{x^2} = 3 + \frac{5}{x^2}$ . Denominator: $\frac{x^2-1}{x^2} = 1 - \frac{1}{x^2}$ .
3
As $x \to \infty$ , $\frac{5}{x^2} \to 0$ and $\frac{1}{x^2} \to 0$ .
4
Limit: $\frac{3 + 0}{1 - 0} = 3$ .

For rational functions at infinity, divide by the highest power of

x

in the denominator. Terms with

x

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 \to \infty} \frac{2 x^{3} - x}{5 x^{2} + 3}

Example 3

medium

Evaluate

\lim_{x \to \infty}(\sqrt{x^2 + 3x} - x)

Example 4

medium

Why is

0 \cdot \infty

an indeterminate form? Give two limits that show different results.

Example 5

hard

Evaluate

\lim_{x \to \infty}\left(\frac{x+2}{x-1}\right)^x

Example 6

challenge

Evaluate

\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 \to \infty} \frac{1}{x^{3}}

Example 2

medium

Evaluate

\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

x

grows larger and larger without bound, what happens to

f(x)=x^2

Example 5

easy

What is

\lim_{x\to0^+}\frac{1}{x}

(as

x

approaches 0 from the right)?

Example 6

easy

What is

\lim_{x\to0^-}\frac{1}{x}

(from the left)?

Example 7

easy

Does

\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

n\to\infty

, what does the sequence

a_n=\frac{1}{n}

approach?

Example 11

medium

Evaluate

\lim_{x\to\infty}\frac{3x^2+5}{x^2-2}

Example 12

medium

Evaluate

\lim_{x\to\infty}\big(\sqrt{x^2+x}-x\big)

Example 13

medium

The geometric series

\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots

has infinitely many terms. What is its sum?

Example 14

medium

Explain why

0\cdot\infty

is called an indeterminate form. Give two products that show different results.

Example 15

medium

Evaluate

\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\to\infty}\frac{\ln x}{x}=0

, explaining what it says about growth rates.

Example 18

challenge

Determine

\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\to\infty}\frac{5x^3-x}{2x^3+7}

Example 21

medium

Evaluate

\lim_{x\to\infty}\frac{x^2+1}{x+3}

Example 22

medium

n\to\infty

, does

a_n=\frac{(-1)^n}{n}

f(x) = \frac{\sqrt{4x^2 + 1}}{x + 3}

x \to +\infty

Example 37

hard

Find

\lim_{x \to -\infty} \frac{\sqrt{4x^2 + 1}}{x + 3}

Example 38

challenge

Evaluate

\lim_{x \to 0^+} x \ln x

Example 39

challenge

Evaluate

\lim_{x \to \infty} \frac{e^x}{x^{10}}

← Back to Infinity Formula Explained Practice Mode

Related Concepts

Limit Asymptote Infinite Geometric Series

Background Knowledge

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

limit