# One-sided limit

A limit where x approaches a from only one side: from the left (x below a) or from the right (x above a). The ordinary limit exists only if both agree.

Maths (university) · Part 02: Limits and continuity · Card 2 of 9 · Last updated 2026-10-19

## Analogy

Two people walking to the same bridge from opposite banks. They meet only if both reach the same spot.

## Example

**f(x) = |x| / x near 0**

For x > 0: f(x) = x/x = 1.
For x < 0: f(x) = -x/x = -1.
Right-hand limit is 1, left-hand limit is -1.
They differ, so no limit at 0.

## In practice

Needed for jumps, absolute values and piecewise functions, where the two sides behave differently.

## Practice

### Flashcards

- **When does a two-sided limit exist in terms of one-sided limits?** When the left-hand and right-hand limits exist and are equal.

## Questions people ask

### How do I write a one-sided limit?

Put a small plus or minus after the number. Write x approaching a from the right as x tends to a plus, and from the left as x tends to a minus. The two-sided limit exists exactly when both one-sided limits exist and are equal to each other.

### When do I need one-sided limits?

Use them for piecewise functions, absolute values, floor functions and square roots near the edge of their domain. For example, the square root of x only has a right-hand limit at 0, since negative inputs are not allowed in the real numbers.

## Related concepts

- [Limit](https://studyvisually.com/learn/science/maths-university/maths-university-part-2/maths-university-part-2-concept-1)
- [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)
- [Limit at infinity](https://studyvisually.com/learn/science/maths-university/maths-university-part-2/maths-university-part-2-concept-5)

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