Practice Infinity in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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.

Showing a random 20 of 50 problems.

Example 1

easy
True or false: lim⁡x→∞sin⁡(x)\lim_{x \to \infty} \sin(x) exists.

Example 2

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

Example 3

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

Example 4

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

Example 5

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

Example 6

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

Example 7

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

Example 8

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

Example 9

hard
Does the series ∑n=1∞1n2\sum_{n=1}^{\infty} \frac{1}{n^2} converge or diverge?

Example 10

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 11

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

Example 12

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

Example 13

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

Example 14

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

Example 15

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

Example 16

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

Example 17

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

Example 18

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

Example 19

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

Example 20

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