# Why does the following limit not exist?

The limit is $$\lim\limits_{(x,y)\to(0,0)}\frac{\sqrt y}{\sqrt x}$$ I thought it was 1.

Also what about $$\lim\limits_{(x,y)\to(0,0)}\frac{ y}{ x}$$

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undefined for negative – evil999man Apr 5 '14 at 7:53
Complex numbers exist then. – Superbus Apr 5 '14 at 7:55

Let $(y_n)=(\frac{1}{n})$ and $(x_n)=(\frac{1}{n^2})$. Then evaluate $\lim \limits_{n\to \infty}\frac{y_n}{x_n}$.
If any of the limits existed, then all the sublimits (for each case) would coincide. So take the limits along the parabolas $y=kx^2$ to find different sublimits.
Note that even if all the limits $y = kx$ coincide, they may still not exist (if you instead approach by $y = k x^2$...). This is a method to disprove, not prove. – Davidmh Apr 5 '14 at 15:30
What I meant is that, if from different paths you get different results, then it does not exist; but you cannot try all possible ways of getting there. There is at least one example where $y=kx$ converges, but $y=kx^2$ doesn't; and I am sure there are the oposite too... or cases where both converge but the limit still does not exist. The core of the issue is that you cannot try all possible ways of approaching your point. – Davidmh Apr 5 '14 at 21:31