How do you find the vertical, horizontal or slant asymptotes for #(5e^x)/((e^x)-6)#?

1 Answer
Feb 11, 2017

Answer:

The vertical asymptote is #x=ln6#
The horizontal asymptote is #y=5# when #x in ]ln6, +oo[#
The horizontal asymptote is #y=0# when #x in ]-oo, ln6[#
No slant asymptote

Explanation:

Let #f(x)=(5e^x)/(e^x-6)#

The domain of #f(x)# is #D_f(x)=RR-{ln6}#

As you cannot divide by #0#, #e^x!=6#

The vertical asymptote is #x=ln6#

As the degree of the numerator is #=# to the degree of the denominator, there is no slant asymptote.

#lim_(x->+oo)f(x)=lim_(x->+oo)(5e^x)/e^x=5#

#lim_(x->-oo)f(x)=lim_(x->-oo)(5)/(1-6/e^x)=0^-#

The horizontal asymptote is #y=5# when #x in ]ln6, +oo[#

The horizontal asymptote is #y=0# when #x in ]-oo, ln6[#

graph{(5e^x)/(e^x-6) [-36.53, 36.54, -18.27, 18.28]}