Find Limit : $\lim_{x\to-∞}\frac {1}{xe^{x}}$

Question

If I'm doing everything correctly, I get to 1/(-∞×0) where -∞×0 is undefined, so I don't know what to do. Also, I cannot do a series expansion because x is approaching infinity.

Can someone please help me?

Answer 1

using l'Hopital Rule

Let :$$ t = -x \implies t \to +∞$$ $$\lim_{x\to-∞}\frac {1}{xe^{x}}=\lim_{t\to＋∞}\frac {1}{-te^{－t}}＝\lim_{t\to＋∞}－\frac {e^{t}}{t}=\lim_{t\to＋∞}－e^{t}=-∞$$