How to find the asymptotes of #f(x)= (3e^(x))/(2-2e^(x))#?

Question

Liam Brennan · Accepted Answer

Horizontal asymptotes #y=0# and #y=1.5# and vertical asymptote #x=0#

One can find asymptotes of #f(x)# two ways

one , finding how #f(x)# behaves or tends as #x->oo# or #x->-oo#

and here as #f(x)=(3e^x)/(2-2e^x)=3/(2e^(-x)-2)#

as #x->-oo#, #e^x->0# and #f(x)->0/2=0#.

Similarly as #x->oo#, #e^(-x)->0# and #f(x)->3/2=1.5#

This gives us horizontal asymptotes as #y=0# and #y=1.5#

two as #f(x)=oo#, #2-2e^x=0# or #e^x=1# i.e. #x=0#

Hence we have a vertical asymptote #x=0#

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

Sadie Alexander · Answer

undefined To find the asymptotes of $ f(x) = \frac{3e^x}{2 - 2e^x} $:

1. Identify the values of $ x $ for which the function is undefined. These are potential vertical asymptotes.

2. Determine the behavior of the function as $ x $ approaches these values from both sides. This can help identify vertical asymptotes.

3. Simplify the function and look for any horizontal asymptotes.

4. Determine the behavior of the function as $ x $ approaches positive and negative infinity. This can help identify horizontal asymptotes.

5. If there are any oblique (slant) asymptotes, find them by dividing the numerator by the denominator using long division or polynomial division.

6. Once you have identified all potential asymptotes, verify them by graphing the function.

Following these steps will help in finding the asymptotes of the given function.