functions with the properties: $f(x) \rightarrow x$ when $x\rightarrow 0$ and $f(x) \rightarrow \frac{1}{x}$ when $x \rightarrow \infty$ I want to find some functions which have both the following asymptotic behaviors:
$f(x) \rightarrow x$ when $x\rightarrow 0$  and $f(x) \rightarrow \frac{1}{x}$ when $x\rightarrow \infty$. I know that $f(x)=\frac{x}{x^2+1}$ has these properties. Are there other functions? Certainly, the simpler the function, the better it is.

Another pretty simple function is
$$
\frac{\tanh^2(x)}x
$$
