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 is a concept about behavior, not a value you can reach or calculate with.

Common stuck point: \infty - \infty is undefined, not 0. \frac{\infty}{\infty} is undefined. These are 'indeterminate forms.'

Sense of Study hint: Substitute a very large number (like 1000 or 1000000) to see whether the expression grows, shrinks, or stabilizes.

Worked Examples

Example 1

easy
Evaluate \lim_{x \to \infty} \frac{3x^2 + 5}{x^2 - 1}.

Solution

  1. 1
    Divide numerator and denominator by the highest power of x in the denominator, x^2.
  2. 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. 3
    As x \to \infty, \frac{5}{x^2} \to 0 and \frac{1}{x^2} \to 0.
  4. 4
    Limit: \frac{3 + 0}{1 - 0} = 3.

Answer

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{2x^3 - x}{5x^2 + 3}.

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}.
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Background Knowledge

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

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