Question

What is the limit of `f(x) = (1 + 4e^-x )/( 3 - 2e^-x)` as x goes to infinity?

Answer 1

`lim_(xrarroo)(1 + 4e^-x )/( 3 - 2e^-x) = 1/3`

Explanation:

As `xrarroo`, we have `e^-x rarr 0`, so `1+4e^-x rarr 1` while `3-2e^-x rarr 3`.

Interestingly, as `xrarr-oo` we get a different limit. We have
`lim_(xrarr-oo)(1 + 4e^-x )/( 3 - 2e^-x) = 4/-2 =-2`