Decrease size by a smaller and smaller value

Question

I'm looking for a function to decrease a scale in size but to a certain smallest size which it can't practically reach. So at first the number is 1. The next step might get to 0.9 and the next 0.85 and so on. What's the best way to achieve this?

Is perhaps log(i) appropriate in some form?

Answer 1

Suppose you want to decrease the value of $\alpha$ upto $\beta$. The following function will work.

$$y=\beta \cdot \left(1+\left(\frac{\alpha}{\beta}-1\right)a^{-bx}\right)$$

Here $a,b \in \mathbb{ R^+}$ are fixed constants with $a>1$.

Answer 2

One choice would be $\frac 12 + \frac 1{2n}$ which starts at $1$ and decreases toward $\frac 12$. It should be clear how to choose the constants to get the start and asymptote you want.