# Limit laws

Limits of sums, differences, products and quotients behave as you would hope: take each limit, then combine. Quotients need a bottom limit that is not zero.

Maths (university) · Part 02: Limits and continuity · Card 3 of 9 · Last updated 2026-10-19

## Analogy

Like adding prices at a till: total the pieces, then combine.

## Example

**Limit of 3x^2 + 1 at x = 2**

Limit of 3x^2 is 3 x 2^2 = 12.
Limit of 1 is 1.
Sum rule: 12 + 1 = 13.
Polynomials: just substitute x = 2.

## In practice

They let you evaluate most limits of polynomials by plain substitution.

## Practice

### Flashcards

- **What is lim of 3x^2 + 1 as x tends to 2?** 13, by substituting x = 2.
- **What does 0/0 tell you about a limit?** It is indeterminate: simplify first and try again.

### 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

### Can I just substitute the value into the function?

For polynomials and other continuous functions, yes. If substituting gives a normal number, that is the limit. If you get 0/0, the rule does not decide the answer and you must simplify first, for example by factorising and cancelling common factors.

### What can I do when substitution gives 0/0?

The form 0/0 is indeterminate, meaning the limit could be anything. Try factorising and cancelling, multiplying by a conjugate, or using L'Hopital's rule later. After cancelling, substitute again. For (x^2 - 9)/(x - 3) at 3, cancel to x + 3, giving 6.

## 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)
- [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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