# Limit at infinity

The value a function approaches as x grows without bound. For fractions of polynomials, divide top and bottom by the highest power of x.

Maths (university) · Part 02: Limits and continuity · Card 5 of 9 · Last updated 2026-10-19

## Analogy

Watching a car drive away: it looks smaller and smaller and settles towards the horizon.

## Example

**(3x^2 + 1) / (x^2 + 5), x large**

Divide top and bottom by x^2.
(3 + 1/x^2) / (1 + 5/x^2).
As x grows, 1/x^2 tends to 0.
Limit is 3 / 1 = 3.

## In practice

Used to find horizontal asymptotes and to compare how fast different functions grow.

## Practice

### Flashcards

- **What is the limit of (3x^2 + 1)/(x^2 + 5) as x tends to infinity?** 3, the ratio of the leading coefficients.

## Questions people ask

### How do I find the limit of a fraction as x goes to infinity?

Divide every term by the highest power of x in the denominator. Terms like 1/x and 1/x squared then vanish. If top and bottom have the same degree, you get the ratio of leading coefficients. If the top has higher degree, the limit is infinite.

### What is a horizontal asymptote?

It is a horizontal line y = L that the graph approaches as x goes to plus or minus infinity. It comes from a limit at infinity. For (3x^2 + 1)/(x^2 + 5), the limit is 3, so the line y = 3 is a horizontal asymptote.

## Related concepts

- [Limit](https://studyvisually.com/learn/science/maths-university/maths-university-part-2/maths-university-part-2-concept-1)
- [One-sided limit](https://studyvisually.com/learn/science/maths-university/maths-university-part-2/maths-university-part-2-concept-2)
- [Limit laws](https://studyvisually.com/learn/science/maths-university/maths-university-part-2/maths-university-part-2-concept-3)
- [Squeeze theorem](https://studyvisually.com/learn/science/maths-university/maths-university-part-2/maths-university-part-2-concept-4)

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