<s>[INST] solve $\lim_{x\to\infty}{\sqrt{x^4+x^2}+\sqrt{x^2+5x}-x^2-x}$ I tried solve this limit.
$\lim_{x\to\infty}{\sqrt{x^4+x^2}+\sqrt{x^2+5x}-x^2-x}$
I tried multiply by $\sqrt{x^4+x^2}-\sqrt{x^2+5x}$, and apply L'Hospital. but this led to alot of work.. and this question seems to had a very easy and fast way to do. I know the answer is 3.
thanks any help.
 [/INST] Continue with ajotatxe's hint: 
$$\lim_{x \to \infty} \sqrt{x^4 + x^2} - x^2$$
$$\lim_{x \to \infty} x^2 (\sqrt{1 + \frac{1}{x^2}} - 1)$$
$$\lim_{x \to \infty} \frac{x^2 (\sqrt{1 + \frac{1}{x^2}} - 1)(\sqrt{1 + \frac{1}{x^2}} + 1)}{\sqrt{1 + \frac{1}{x^2}} + 1} $$
$$\lim_{x \to \infty} \frac{x^2  \frac{1}{x^2}}{\sqrt{1 + \frac{1}{x^2}} + 1} $$
$$\lim_{x \to \infty} \frac{1}{\sqrt{1 + \frac{1}{x^2}} + 1} $$
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