Determining if a function decreases exponentially

Question

Define a function:

$f(x) = \sqrt{\frac{e^{-kx}}{1-e^{-kx}}}$

where $k > 0$.

Does this function decrease exponentially?

EDIT: Sorry, I meant to ask just if it decreases exponentially.

Answer 1

First note that $\sqrt{e^{-kx}} = e^{-kx/2}$ for the numerator. For the denominator, note that $1-e^{-kx} \leq 1,\,\forall x\geq 0$, and $1-e^{-kx} \geq \frac{1}{4}$ (say) when $x$ is sufficiently large. Hence, $\frac{1}{4} \leq 1-e^{-kx} \leq 1$ when $x$ is sufficiently large, and thus $$e^{-kx/2} \leq \sqrt{\frac{e^{-kx}}{1-e^{-kx}}} \leq 2e^{-kx/2}$$ and your function decays "exactly exponentially."