# $\lim\limits_{} u_n= 0$ and $\lim\limits_{}v_n = 0$ then why $\lim\frac{u_n}{v_n}$ is not defined

by Salutsalut1   Last Updated January 16, 2018 20:20 PM

Let $u_n$ be a function $\mathbb{N} \rightarrow \mathbb{R}$ and $v_n$ be a function $\mathbb{N} \rightarrow \mathbb{R}$ such that : $v_n, u_n \ne 0$ for all $n \in \mathbb{N}$ and such that : $\lim\limits_{} u_n= \lim\limits_{} v_n = 0$.

I would like to understand intuitively why we can't say that : $\lim\limits_{} \frac{u_n}{v_n} = 1$. I know many counter-example why this doesn't work, but I don't understand that intuitively. I mean is there a nice way to explain why this is false ?

Why when $\lim\limits_{} u_n= \lim\limits_{} v_n = r$, $r \in \mathbb{R}^{*}$ we have : $\lim\limits_{} u_n/v_n = 1$, I don't see why it makes a difference in the assessment when $r = 0$.

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The reason is that $\frac{r}{r}=1$ is true only for $r\neq 0$ thus you can calculate the limit by algebraic rules. When $v_n$ and $u_n$ tend to 0 you need to consider “how” they tend to 0 and in this case the limit can assume any value or do not exist at all.