Hints about the limit $\lim_{x \to \infty} ((1+x^2)/(x+x^2))^{2x}$ without l'Hôpital's rule? I've tried to evaluate $\lim_{x \to \infty} \left(\frac {1+x^2}{x+x^2}\right)^{2x}$ as $$\lim_{x \to \infty} \left(\left(\frac {1+ \frac{1}{x^2}}{1+ \frac{1}{x}}\right)^{x}\right)^{2}$$
So the denominator goes to $e^2$, but I don't know how to solve the numerator, because of the $x^2$. Any hint?
Thanks in advance!

$\lim\limits_{x \to \infty} \ln\left(\frac {1+x^2}{x+x^2}\right)^{2x}=\lim\limits_{x \to \infty}2x\ln(1+\frac{1-x}{x+x^2})=\lim\limits_{x \to \infty}2x\frac{1-x}{x+x^2}=-2$
So $\lim\limits_{x \to \infty}\left(\frac {1+x^2}{x+x^2}\right)^{2x}=e^{-2}$
