Question

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

Answer 1

There is no vertical asymptote. (assuming we are restricted to the Real number plane)
Horizontal asymptotes at `y=1` and `y=0`

Explanation:

Vertical Asymptote
Since `e^x > 0` for all Real values of `x`
the denominator of `(e^x)/(1+e^x)` will never be `=0`
and the expression is defined for all values of `x`

Horizontal Asymptote
`(e^x)/(1+e^x) = ((e^x)/(e^x))/(1/(e^x)+(e^x)/(e^x)) = 1/(1/(e^x)+1)`
and
since
`color(white)("XXX")1/(e^x)rarr 0` as `xrarr+oo`
and
`color(white)("XXX")(e^x)rarr0` as `xrarr-oo`
`color(white)("XXX")`which implies `1/e^x rarr oo` as `xrarr -oo`

therefore
`color(white)("XXX")lim_(xrarr+oo) y = 1/(0+1) = 1`
and
`color(white)("XXX")lim_(xrarr-oo) y = 1/(oo+1) = 0`

For verification purposes, here's what the graph looks like:
graph{e^x/(1+e^x) [-6.24, 6.244, -3.12, 3.12]}