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) exists.

Example 2

medium
As n→∞, does an=(−1)nn converge? To what?

Example 3

easy
Is the expression ∞∞ automatically equal to 1?

Example 4

medium
Evaluate lim⁡x→∞(x2+3x−x).

Example 5

medium
Find the horizontal asymptote of f(x)=3x2−1x2+5.

Example 6

medium
Evaluate lim⁡x→∞(x2+x−x).

Example 7

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

Example 8

easy
Evaluate lim⁡x→∞3x2+5x2−1.

Example 9

hard
Does the series ∑n=1∞1n2 converge or diverge?

Example 10

medium
The geometric series 12+14+18+⋯ has infinitely many terms. What is its sum?

Example 11

challenge
Evaluate lim⁡x→0+xln⁡x.

Example 12

medium
Evaluate lim⁡x→1−1x−1.

Example 13

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

Example 14

easy
Evaluate lim⁡x→∞e−x.

Example 15

medium
Evaluate lim⁡x→∞5x3−x2x3+7.

Example 16

easy
Evaluate lim⁡x→∞(7−2x).

Example 17

medium
Evaluate lim⁡x→∞4x3+2xx3−x2+1.

Example 18

easy
As n→∞, what does the sequence an=1n approach?

Example 19

medium
Evaluate lim⁡n→∞(1−1n)n.

Example 20

easy
Does lim⁡x→01x2 exist as a finite number?
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