How do you find the Vertical, Horizontal, and Oblique Asymptote given #(6e^x)/(e^x-8)#?

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Answer 1 · 2016-12-01T06:20:24

The vertical asymptote is #x=ln8#
The horizontal asymptotes are #y=0# and #y=6#
No oblique asymptote

As you cannot divide by #0#

The denominator must be #!=0#

#e^x-8!=0#

#e^x!=8#

#x!=ln8#

So the vertical asymptote is #x=ln8#

Let #f(x)=(6e^x)/(e^x-8)#

#f(x)=(6e^x)/(e^x-8)=6/(1-8e^(-x))#

#lim_(x->-oo)f(x)=lim_(x->-oo)6/((1-8e^(-x)))=6/-oo=0^(-)#

#lim_(x->+oo)f(x)=lim_(x->+oo)6/((1-8e^(-x)))=6#

The horizontal asymptotes are #y=0# and #y=6#

graph{(6e^x)/(e^x-8) [-15.04, 16.99, -5.05, 10.96]}