You could convert to polars...
Remember that , then
.
Find if it exists.
After testing different paths, the limit appears to be 0, so let's try an proof. I got a bit rusty at this, I just don't know how to finish it off lol.
We need to prove if for every there exists a such that if then
We know that
But
Thus we can take but then how do we relate this back to again? I forgot lols
The best way to handle a problem like this is to change to polar coordinates. That way, only the single variable, r, measure how "close" to (0, 0) you are. If the limit as r goes to 0 exists and is the same for all , then the limit itself exists.
Of course, , , and so this problem becomes
for all .
If you want to do a strict " " proof just do it as you would for one variable with r and leave alone.
(You can't take " " for two reasons:
1) is given- it can be anything. It is you are to "take".
2) You must get in terms of , not x and y. It must be a constant, not a function of x and y.)
Usagi, you would take [LaTeX ERROR: Convert failed] and I think you'll be correct.
Halls, Prove, here is an example from my textbook that may trouble you
Let [LaTeX ERROR: Convert failed]
With your method, set and . Then the new function is
Taking the limit as r approaches 0, we get 0.
HOWEVER, let . Now we have
for all values.
Hence the limit does not exist and your method fails!