How do you find the vertical, horizontal or slant asymptotes for #y= x / e^x#?

Question

1367 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-11-16T15:24:57

The horizontal asymptote is #y=0#
No slant or vertical asymptote.

The exponential function #e^x# is always positive.

#AA x inRR, e^x>0#

So we don't have a vertical asymptote:

#lim_(x->-oo)y=x/e^(-oo)=(-oo*e^(oo))=-oo#

#lim_(x->+oo)y=x/e^(+oo)=(x/e^(oo))=0^(+)#

We have a horizontal asymptote #y=0#

graph{(y-x/e^x)(y)=0 [-6.29, 7.757, -4.887, 2.136]}