Algebra of limits

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, limit of a product of sequences and limit of a quotient of sequences. Similar rules hold for the limit of a function.

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 $(2 + 0)/(1 + 0) = 2$.