From this expression ax/(b+cx), how can I have x only in the numerator?

Question

I have an expression ax/(b+cx) where a, b and c are known but I need to get an expression like this (A/B)* x where A, B may be a, b and c manipulated to get (A/B) x.

Does anybody know how to do this?

Answer 1

You can't because $(A/B)x$ gets large in absolute value for large $x$ (unless $A=0$) while

$\begin{array}\\ \frac{x}{b+cx} &=\frac1{c}\frac{cx}{b+cx}\\ &=\frac1{c}\frac{cx+b-b}{b+cx}\\ &=\frac1{c}(1-\frac{b}{b+cx})\\ &=\frac1{c}-\frac{b}{c(b+cx)}\\ \end{array} $

is bounded for large $x$.