What is a slower decay rate than hyperbolic?

Question

Hyperbolic decay

$$f_\alpha(x)=\frac{1}{\alpha x + 1} $$

is slower than exponential decay

$$f(x) = e^{-x}$$

where $\alpha > 0$ is a scaling factor. The larger that $\alpha$ is, the steeper the descent towards $0$.

What would be a slower decay function than hyperbolic?

Answer 1

Any reciprocal of a function that grows slower than linear. E. g. a much slower decay rate than hyperbolic would be $$\frac{1}{\ln(x)}$$ or even $$\frac{1}{\ln(\ln(x))}$$

Answer 2

$$f(x) = \frac{1}{x}.$$

Reciprical square root,

$$g(x)=\frac{1}{\sqrt{x}} $$

or $x^{-p}$ for any $0<p<1$:

$$h_p(x)=\frac{1}{x^p}. $$

These are much slower than $\frac{1}{x}$ as $x \to \infty$, since

$\large{\frac{f(x)}{h_p(x)} = \frac{\frac{1}{x}}{\frac{1}{x^p}} = \frac{x^p}{x} =}$ $x^{p-1}\to 0 $ as $ x \to \infty\ $ because $p-1<0. $