Solving this limit without l'Hôpital $\lim_{x \to -\infty} \sqrt{x^2 +x + 4} + x$ I'm trying to solve this limit $\ \lim_{x \to -\infty} \sqrt{x^2 +x + 4} + x $. I tried solving it by calculating the limit of $\sqrt{x^2 +x + 4}$ and $ \ x $ in apart from eachother and simply adding them, however this would get me 0 and wolfram doesn't seem to agree with me on that one. I also tried to find this limit by multiplying by it's conjugate expression which ended up with me trying to divide by zero, so I doubt I found the solution there. Any help would be greatly appreciated

Set $y=-1/x$ to get $$x^2+x+4=\dfrac{1-y+4y^2}{y^2}$$
$$\sqrt{x^2+x+4}=+\dfrac{\sqrt{1-y+4y^2}}{|y|}$$ 
as $y>0, |y|=+y$
$$\lim_{x\to-\infty}\sqrt{x^2+x+4}-x=\lim_{y\to0^+}\dfrac{\sqrt{1-y+4y^2}-1}y$$
Now rationalize the numerator
