---
tag: 07XBM3A9J5
kind: concept
name: Algebra of limits
aliases: [limit-laws]
msc: [40A05]
uses: [0XKWSSY8P2]
sources:
  - url: https://en.wikipedia.org/wiki/Limit_of_a_sequence
    locator: Properties
---

## Summary

The algebra of limits is a set of rules for limits of convergent sequences.
The rules give the limit of a sum, a product or a quotient from the limits of the parts. If $a_n \to a$ and $b_n \to b$, then $a_n + b_n \to a + b$, $a_n b_n \to a b$, and $a_n / b_n \to a / b$ when $b \ne 0$. These rules let us find many limits from a few known ones.

## Remarks

The rules are the nodes [[limit of a sum of sequences|0RAM4E07NR]], [[limit of a product of sequences|0734RSVF2V]] and [[limit of a quotient of sequences|03S6GWWKEZ]].
Similar rules hold for the [[limit of a function|0TXTSEMHR5]].

## Explanations

### L1 human

To find the limit of $(2n + 1)/(n + 3)$, divide the numerator and the denominator by $n$ to get $(2 + 1/n)/(1 + 3/n)$.
Since $1/n \to 0$, the rules give the limit $(2 + 0)/(1 + 0) = 2$.