How do you find the horizontal asymptote for ##f(x)= (3e^(x))/(2-2e^(x))##?

##y=0## ##y=-3/2##

To find any horizontal we have to evaluate:

##lim_(x rarr+-oo)f(x)##

if the limits are finite then they are horizontal asymptotes

##lim_(x rarr-oo)f(x)=lim_(x rarr-oo)3/2(e^x/(1-e^x))=##

##3/2lim_(x rarr-oo)(e^x/(1-e^x))~~3/2lim_(x rarr-oo)e^x/1=3/2*0=0##

##:. y=0## is a horizontal asymptote for ##x rarr-oo##

##lim_(x rarr+oo)f(x)=lim_(x rarr+oo)3/2(e^x/(1-e^x))=##

##3/2lim_(x rarr+oo)(e^x/(1-e^x))~~3/2lim_(x rarr+oo)(e^x/-e^x)=##

##=3/2*(-1)=-3/2##

##:. y=-3/2## is a horizontal asymptote for ##x rarr+oo##

graph{(3/2)*((e^x)/(1-e^x)) [-10, 10, -5, 5]}