Question: Let $x_n = \frac 1{n^2}$ and $y_n = \frac 1n$. Define $z_n = \frac {x_n}{y_n}$ and $w_n = \frac {y_n}{x_n}$. Do $z_n$ and $w_n$ converge? What are the limits? Can you apply Proposition 2.2.5? Why or Why not?
From the question, we can see that $\lim x_n = 0$ and $\lim y_n =0$. Therefore, we cannot use the proposition 2.2.5, which is $\lim \frac {x_n}{y_n} = \lim x_n {\frac 1{\lim y_n}}$, because 1 cannot be divided by 0.
This is the only way that I know to find the limit of fraction sequence. To find the limit and convergence in this question, how should I approach it? Could you give some hint?
Thank you in advance.
... so $z_n = 1/n$ and $w_n = n$ ... Don't forget to substitute in the definition of $z_n$ and $w_n$ before taking the limit (since doing it in that order is literally what the notation is telling you to do).