TITLE: Why does the following limit not exist?
QUESTION [6 upvotes]: The limit is $$\lim\limits_{(x,y)\to(0,0)}\frac{\sqrt y}{\sqrt x}$$  I thought it was 1.
Also what about $$\lim\limits_{(x,y)\to(0,0)}\frac{ y}{ x}$$

REPLY [11 votes]: Let $(y_n)=(\frac{1}{n})$ and $(x_n)=(\frac{1}{n^2})$. Then evaluate $\lim \limits_{n\to \infty}\frac{y_n}{x_n}$.