two variable limit question

[sonic grail]

2*x*y / (x**2 - y + 5*x*y)
how can i prove that this function has not limit at the point (0,0)

i have tried polar coordinates and y=k*x, y=k*x**2,
y = k*x**3 and y=k*x**0.5 but all of them resulted with 0 so i thought it is 0 but calculator says no

[viral karma]

Hello! could you show your work?

Maybe when you tried polar coordinates, or y=kx, if possible

[sonic grail]

i am trying to solve many questions in my notebook and it will seem complicated, but if you want to see i can rewrite clearly to a blank page

[viral karma]

Sure

Maybe you got 0/... , in that case you should check that the denominator can't be 0

[sonic grail]

ok i will send

if k != 0 the answer should be 0 right?

[viral karma]

Yes, but since there's a value of k (path) where the limit doesn't exist, then the whole limit doesn't exist

For the limit to exist, you must get the same finite value for all k, theta, etc. depending on the method you use

[sonic grail]

ohh i see because of k could be 0 the limit will in danger if not limit is 0, so this dilemma makes our limit dependent