8: Limit at −∞:

If you tried to use l'Hospital's rule right away, you noticed that it did did not help. There are two reasonable ways to arrive at that zero. The first is to cancel e^x in the fraction and then use l'Hospital's rule, it does help now. The second possibility is to use l'Hospital's rule to prove that xe^x goes to zero at minus infinity (it must be changed into a fraction to be eligible for l'H).
Horizontal asymptote y = 0 at −∞.

Limit at x = 0 from the left:

No vertical asymptote at x = 0 yet.

Limit at x = 0 from the right:

No vertical asymptote at x = 0 confirmed.

Limit at ∞:

No horizontal asymptote at ∞, but a chance for oblique.

Oblique asymptote y = x at ∞.

Answer