Looking for functions

Question

I am looking for functions that

• have a minimum at 0 [edit: I meant f'(0)=0]
• and are asymptotically linear

I came up with $\tanh(x) x$, which is fine, but more suggestions would be welcome. (It's needed for data-fitting.)

Thx in advance, Jo

Answer 1

You should declare the domain in question, for example: $f: \mathbb R_{+} \to \mathbb R$ given by $f(x)=x$ works.

Otherwise, consider $g: \mathbb R \to \mathbb R$ given by $g(x)=\frac{x^2}{x+1}$.

I thought of this by noticing that for positive $x$, $\frac{x^2-1}{x+1} \leq g(x) \leq \frac{x^2+1}{x+1}$.

Either way, to really see it, take $h(x)=x$ and, so $\lim_{x \to \infty}\frac{g(x)}{h(x)}=\lim_{x \to \infty}\frac{x}{x+1}=1.$

Answer 2

$$\quad\quad\quad\quad\quad f(x)=C|x|+g(x)$$

where $\text{argmin}(g(x))=0$ and $g(x)=o(x)$ for $x\rightarrow \infty$. For example $g(x)=0,\frac{1}{1+x^2},e^{-|x|}...$