# Quotient Rule for Sequences

## Contents

## Theorem

### Quotient Rule for Real Sequences

Let $\sequence {x_n}$ and $\sequence {y_n}$ be sequences in $\R$.

Let $\sequence {x_n}$ and $\sequence {y_n}$ be convergent to the following limits:

- $\displaystyle \lim_{n \mathop \to \infty} x_n = l$
- $\displaystyle \lim_{n \mathop \to \infty} y_n = m$

Then:

- $\displaystyle \lim_{n \mathop \to \infty} \frac {x_n} {y_n} = \frac l m$

provided that $m \ne 0$.

### Quotient Rule for Complex Sequences

Let $\sequence {z_n}$ and $\sequence {w_n}$ be sequences in $\C$.

Let $\sequence {z_n}$ and $\sequence {w_n}$ be convergent to the following limits:

- $\displaystyle \lim_{n \mathop \to \infty} z_n = c$
- $\displaystyle \lim_{n \mathop \to \infty} w_n = d$

Then:

- $\displaystyle \lim_{n \mathop \to \infty} \frac {z_n} {w_n} = \frac c d$

provided that $d \ne 0$.

## Also presented as

Some treatments of this subject specifically exclude all sequences where the denominators are zero at **any** point in their domain.

Thus, for example, this is how it is presented in 1960: Walter Ledermann: *Complex Numbers*:

*If $z_n \to c$ and $w_m \to d$, then**... $\text{(iv)} \ z_n / w_n \to c / d$, where ... $w_n \ne 0$ for all $n$ and $d \ne 0$.*

However, it is demonstrated within the proof that past a certain $N \in \R$, which is bound to exist, $w_n$ is *guaranteed* to be non-zero.

The behaviour of the sequence $S = \sequence {\dfrac {z_n} {w_n} }$ in the limit is not dependent upon the existence or otherwise of its terms for $n < N$.

Thus it is not necessary to state that $w_n \ne 0$ for all $n$, and in fact such a statement would unnecessarily restrict the applicability of the theorem to exclude otherwise well-behaved cases where it is desirable that the theorem *does* apply.

Hence this restriction is not supported on $\mathsf{Pr} \infty \mathsf{fWiki}$.