# Limit

The limit of f(x) as x approaches a is the value f(x) gets closer and closer to. The function does not need to be defined at a itself.

Maths (university) · Part 02: Limits and continuity · Card 1 of 9 · Last updated 2026-10-19

## Analogy

Walking towards a door: you judge where it is by your steps getting near, even before you touch it.

## Example

**Limit of (x^2-4)/(x-2) at 2**

At x = 2 this is 0/0, undefined.
For x not equal 2: (x-2)(x+2)/(x-2) = x + 2.
As x approaches 2, x + 2 approaches 4.
So the limit is 4.

## In practice

Limits let you handle 0/0 situations, which is how derivatives are built.

## Practice

### Flashcards

- **What is lim of (x^2 - 4)/(x - 2) as x tends to 2?** 4. Cancel (x - 2) to get x + 2.

### Interview questions

**Evaluate the limit of (x^2 - 9)/(x - 3) as x tends to 3 and explain each step.**

- Substituting gives 0/0, which is indeterminate
- Factorise the top as (x - 3)(x + 3)
- Cancel the common factor for x not equal 3
- Substitute x = 3 into x + 3 to get 6

## Questions people ask

### Does a limit have to equal the function's value?

No. The limit describes what happens near a, not at a. The function may be undefined there, or have a different value. When the limit does equal f(a), the function is called continuous at a. Otherwise there is a gap, jump or hole in the graph.

### What does it mean when a limit does not exist?

The values of f(x) near a do not settle on one number. They may jump between two values, grow without bound, or oscillate wildly. For example, 1/x near 0 grows large on both sides in opposite directions, so no single limit exists.

## Related concepts

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