Evaluate a the following limit in two variables:
$$\lim_{(x,y)\to(0,0)\ }\ \frac{\cos^{2}(x)-1+y^{2}}{\ \ \arctan^{4}(x)+y^{2}}$$ or prove it doesn't exist.
I tried to simplify the expression, by replacing: $\cos^{2}(x) -1= \sin^{2}(x)$. After that I was stuck.
Also, I know it's possible to find some limit under the assumption that the limit exists. But if I'm no allowed to make such assumption, is it allowed to calculate the limit by fixing the value of $x$ and then fixing the value of $y$?
I can't find a way to prove that this limit doesn't exist.
We have:
For $x=0,$ $$\frac{\cos^{2}(0)-1+y^{2}}{\arctan^{4}(0)+y^{2}}=\frac{y^2}{y^2}=1$$
For $y=0,$ $$\frac{\cos^{2}(x)-1+0^{2}}{\arctan^{4}(x)+0^{2}}=\frac{\cos^{2}(x)-1}{x^2}\frac{x^4}{\arctan^{4}(x)}\frac1{x^2}\to \infty$$