When does $\lim _{n \rightarrow \infty} \frac {a_n}{b_n} = \frac{\lim_{n \rightarrow \infty} a_n}{\lim _{n \rightarrow \infty}b_n}$.

Question

I need to use property $\lim _{n \rightarrow \infty} \frac {a_n}{b_n} = \frac{\lim_{n \rightarrow \infty} a_n}{\lim _{n \rightarrow \infty}b_n}$ but I am not sure about that.

Could you please tell me when $\lim _{n \rightarrow \infty} \frac {a_n}{b_n} = \frac{\lim_{n \rightarrow \infty} a_n}{\lim _{n \rightarrow \infty}b_n}$?

Is it true that we only need the condition $\lim_{n \rightarrow \infty}b_n \neq 0$?

Thank you for your help.

Answer 1

If $\lim a_n$ and $\lim b_n$ exist and $\lim b_n\ne 0$, then $\lim \frac{a_n}{b_n}$ exists and equals $\frac{\lim a_n}{\lim b_n}$.

(Aka: division is continuous on all of its domain)

Answer 2

We require that both limits exist and the limit in the denominator not to be zero.

In general, if you wish to use continuity properties, you'll need the limits on both sides to exist plus any additional restrictions that make sense.

Answer 3

We know that if $a_{n}\rightarrow a$ and $c_{n}\rightarrow c$, then $a_{n}c_{n}\rightarrow ac$.

Now, suppose you have a sequence $b_{n}\rightarrow b\neq0$. Since $b\neq0$, we can define the sequence $c_{n}=1/b_{n}$ (for $n$ sufficiently large, $b_n \neq 0$). Note that $c_{n}\rightarrow 1/b$ since $x\mapsto1/x$ is continuous away from zero.

Now, can you combine the above two facts to arrive at the desired result?