# How do you find the asymptotes for # f(x)=(e^x)/(1+e^x)#?

##### 1 Answer

There are no vertical asymptotes, and two horizontal asymptotes at

#### Explanation:

Vertical asymptotes of a rational function such as this one could occur where the function's denominator equals

Setting the denominator equal to

So, we've determined there are no vertical asymptotes. However, this doesn't account for horizontal asymptotes. We can determine these by finding the limit of the function as it approaches infinity.

Another important thing to remember is that, horizontally, we have *two* infinities, that is, positive and negative infinities. So, we have to take the limit at both of these values.

At positive infinity:

#lim_(xrarroo)e^x/(1+e^x)=lim_(xrarroo)1/(1/e^x+1)=1/(0+1)=1#

So, there is a horizontal asymptote at

At negative infinity, we see that

#lim_(xrarr-oo)e^x/(1+e^x)=0/(1+0)=0#

Our other horizontal asymptote is at

Check a graph of the function!

graph{e^x/(1+e^x) [-13.7, 11.61, -1, 2]}