Limit of a 2-variable function Given problem:
$$
\lim_{x\to0,y\to0}(1+x^2y^2)^{-1/(x^2+y^2)} 
.$$
I tried to do it with assigning y to $y = kx$, but that didn't help me at all. Also one point, I can't use L'Hospital's rule.

The expression equals
$$\tag 1\left ((1+x^2y^2)^{1/(x^2y^2)}\right )^{-x^2y^2/(x^2+y^2)}.$$
Because $(1+u)^{1/u} \to e$ as $u\to 0,$ the expression inside the parentheses $\to e.$ Since
$$0\le x^2y^2/(x^2+y^2) \le x^2[y^2/(x^2+y^2)] \le x^2,$$
the outside exponent in $(1)$ goes to $0.$ The desired limit is therefore $e^0=1.$
(Note: There's a small problem above. What if $x^2y^2=0,$ which would render $1/(x^2y^2)$ meaningless? That's really no problem because the expression of interest actually equals $1$ when $x^2y^2=0.$)
