1. ## limit

Show that the following does not exist
Lim (x,y) to (0.0) 13y^2/ x^4+y^2+x^2

I try to use polar coordinate x=r cos, y=r sin or y=mx but somehow it is working or this equation look like exist.

2. $\displaystyle \displaystyle \lim_{(x,y)\to (0,y)}\frac{13y^2}{y^2}=13$

$\displaystyle \displaystyle \lim_{(x,y)\to (x,0)}\frac{0}{x^4+x^2}=0$

3. If this is the answer, this equation is exist?

4. Do you understand what a limit is? As you get close to the target point, (0, 0), along any path, you must get close to the limit value. That does NOT happen here. You can be arbitrarily close to (0, 0) and have values close to 0 (on the line (x, 0)) or have values close to 13 (on the line (0, y)).

If you convert to polar coordinates you get $\displaystyle \frac{13 r^2 sin^2(\theta)}{r^4 cos^4(\theta)+ r^2}= \frac{sin^(\theta)}{r^2 cos^4(\theta)+ 1}$ and, to have a limit at (0, 0), that must be independent of $\displaystyle \theta$ as r goes to 0. What makes you say "somehow it is working"?