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!

HINT
We have
$$\left(\frac {1+x^2}{x+x^2}\right)^{2x}=\left(\frac {x+x^2+1-x}{x+x^2}\right)^{2x}=\left[\left(1+\frac {1-x}{x+x^2}\right)^{\frac {x+x^2}{1-x}}\right]^{\frac {2x(1-x)}{x+x^2}}$$
