So I have to find the horizontal asympottes of
$$ f(x)=(\sqrt{x^2+1}-|x|)(x+2) $$
Relatively straightforward. For the limit at infinity, the absolute value I turn into a positive and solve it and I get 1/2 which is correct.
But for the negative infinity portion, I get division by 0 if I simplify and "plug in" infinity.
I'm not allowed to use L'Hopital's rule on this. I rationalized it as normal but I end of getting:
$$ \lim_{x \rightarrow \infty}\frac{ x+2 }{ \sqrt{x^2+1}-x } $$
But if I got ahead and simplify that, I get get $$1/0$$ which is clearly undefined. Any ideas?
I used the assumption that I turn the absolute value negative as it goes to negative infinity. Is that wrong?