# Thread: need help generating formula to fit desired results

1. ## need help generating formula to fit desired results

I am unsure if this is the correct forum for my question. However, Calculus 1 is the highest level of math I have completed, and I am able to define the requirements of the formula with limits. Feel free to move the thread if it is in the wrong forum.

I need to define i.
There are three variables: x, y, and z.
However, x and y are only considered in the form (x+2y), so we can replace that with a single variable for simplicity, let's say w.
All of these variables are integers.
I'm fairly certain that i can be defined as i = w/(some expression).
w >= 0
w < z
z >= 1
0 < i < 1

as w -> 0, i -> w/z
as w -> infinity, i -> w/w
Notice that the denominator is the only part that changes, and that it defines i to be between 0 and 1. But I need one function that will work for both limits.

For most circumstances, using i=w/z would work just fine. But when w gets larger, the formula becomes less reliable. As w increases, the z variable increases at a growing ratio. It needs to be a curved relationship, such that values of w < 100 or so still are much closer to using i=w/z. But for values of w > 1000, it should already be very close to i=w/w.

$\displaystyle i = \frac{w}{w + ze^{-w}}$ ?