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.

\begin{align}
\lim_{x\to\infty}{\sqrt{x^4+x^2}+\sqrt{x^2+5x}-x^2-x}
&= \lim_{x\to\infty}\sqrt{\left(x^2+\dfrac12\right)^2-\dfrac14} \\
&+\sqrt{\left(x+\frac52\right)^2-\frac{25}{4}} -x^2-x \\
&= \lim_{x\to\infty}{\left(x^2+\dfrac12\right)+\left|x+\frac52\right|-x^2-x} \\
&= \lim_{x\to+\infty}{\dfrac12+x+\frac52-x} \\
&= \color{blue}{3}
\end{align}
