Why is $\lim_{x\to -\infty}\sqrt{x^2+5x+3}+x = \lim_{x\to -\infty}(-x)+x=\lim_{x\to -\infty}0 = 0$ not correct? $\lim_{x\to -\infty}\sqrt{x^2+5x+3}+x = \lim_{x\to -\infty}(-x)+x=\lim_{x\to -\infty}0 = 0$
Apparently, the 2nd step is illegal here. Probably because for $x=-\infty$ I'd get $(+\infty-\infty)$ which is not possible. I see why this wouldn't be possible, I'm not sure if it really is the cause which makes that equation illegal though.
But now, apparently, I could do this:
$\lim_{x\to -\infty}\sqrt{x^2+5x+3}+x=\lim_{x\to -\infty}\frac{[\sqrt{x^2+5x+3}+x][\sqrt{x^2+5x+3}-x]}{\sqrt{x^2+5x+3}-x}=\lim_{x\to -\infty}\frac{x^2+5x+3-x^2}{\sqrt{x^2+5x+3}-x}=\lim_{x\to -\infty}\frac{5x+3}{\sqrt{x^2+5x+3}-x}=\lim_{x\to -\infty}\frac{5+3/x}{\sqrt{1+5/x+3/x^2}-1}=-5/2$
which gives me the correct result.
But in the 3th step I used $x^2-x^2=0$, how is that legal?
Also, in the 2nd step I implicitly used:
$-x\sqrt{x^2+5x+3}+x\sqrt{x^2+5x+3}=0$
Which also seems to be fine, but why? 


$\lim_{x\to -\infty}\sqrt{x^2+5x+3}+x = \lim_{x\to -\infty}(-x)+x$
Apparently, the 2nd step is illegal here.

In this step, you dropped $5x+3$ to go from $\sqrt{x^2}$ to $-x$, but what makes you think you can just leave out the terms $5x+3$...? Surely you agree that $x^2+5x+3 \ne x^2$.

But in the 3th step I used $x^2-x^2=0$, how is that legal?

How could that not be legal? The difference of two equal real numbers is $0$, of course!
The same goes for:

Also, in the 2nd step I implicitly used:
$-x\sqrt{x^2+5x+3}+x\sqrt{x^2+5x+3}=0$


Addition after comment: the idea of dominating terms (in a polynomial) is fine, but you cannot choose to apply the limit "locally" and leave the other terms unchanged.
To clarify, this is fine:
$$\lim_{x \to -\infty}\sqrt{x^2+5x+3}=\lim_{x \to -\infty}\sqrt{x^2}=+\infty$$
but that's not what we have here; we have:
$$\lim_{x \to -\infty}\left(\sqrt{x^2+5x+3}\color{blue}{+x}\right)$$
and we cannot take the limit of the terms separately
$$\lim_{x \to -\infty}\left(\sqrt{x^2+5x+3}\color{blue}{+x}\right)\color{red}{\ne}\lim_{x \to -\infty}\sqrt{x^2+5x+3}+\lim_{x \to -\infty} x$$
because both limits need to exist in order for this step to be valid.
