Goodmoring, I'm having difficulty in resolving 2 variable limits with some variable substitution. I can't understand when the substitution is legit or not. My calculus teacher told me that I've to substitute x and y with an invertible function in order to not excluding some paths.
For example, i was trying to solve $\lim_{(x,y)->(0,0)} \frac{x^2-2y^2}{x-3y}$. In the first place i tried $y=-\frac{t}{3}$, which is invertible, so the limit becomes $\lim_{(x,t)->(0,0)} \frac{x^2-\frac{2}{9}t^2}{x+t}$.
The limit should not exist(it can be verified taking 2 different curve) but if i try the following substitution(with all invertible functions):
- For $x>0,t>0$: $a=\sqrt{x}, b=\sqrt{t}$
- For $x<0,t>0$: $a=-\sqrt{-x}, b=\sqrt{t}$
- For $x<0,t<0$: $a=-\sqrt{-x}, b=-\sqrt{-t}$
- For $x>0,t<0$: $a=\sqrt{x}, b=-\sqrt{-t}$
The limit in all the cases about becomes $\lim_{(a,b)->(0,0)} \frac{a^4-2b^4}{a^2+b^2}$.
If i substitute $a=\rho cos(\theta), b=\rho sin(\theta)$ the limit becomes: $\lim_{\rho->0} \rho^2(cos^4(\theta)-sin^4(teta))=0$.
So, I don't know what I'm doing wrong(It's my calculus teacher who suggested to split the problem and to do 4 substitution. Is it wrong?)