Finding an example a pair of sequences converging to 0 but such that their quotient diverges

Question

I was tasked with finding an example for a pair of sequences $ (a_{n})_{n=1}^\infty$ , $ (b_{n})_{n=1}^\infty$, such that $\lim_{n\to\infty}b_{n} = \lim_{n\to\infty}a_{n}= 0$, and at the same time $$\lim_{n\to\infty}\frac{a_{n}}{b_{n}}$$

doesn't exist.

In this context, I don't really understand what does ''Not existing'' mean....does it mean the limit goes to ${\infty}$ ? Or rather does it have something to do with partial sums of sub-sequences, such as some $2n$ and $2n-1$ whose limit of a ratio must not exist?

Answer 1

You can take $a_n = 1/n$, and $b_n = 1/n^2$. Then $$\lim_{n\to\infty} \frac{a_n}{b_n} = \lim_{n\to\infty} n,$$ which does not exist.

As for your second question, not existing means that the ratio $a_n/b_n$ does not converge towards a real value. This happens e.g. if the sequence tends to infinity, or if it fluctuates between separate values.

Answer 2

Let $a_n= 1/n$ and $b_n= (-1)^n 1/n$.