Function with special behavior near zero

Question

I'm looking for functions that have the following behaviors:

1. $f(x) \to 0$ as $|x| \to 0$,

2. as $x \to 0^+$, $\alpha < \frac{{df(x)}}{{d(x)}}$ for any $0<\alpha<\infty $.

One example of this kind of functions is $f(x)=x^a$ where $0<a<1$.

Thank you for your help.

Answer 1

1.

$f(x)=-x\ln{|x|}$. Verify:

$f(0^{+})=f(0^{-})=0$, so $\lim_{x\to0}f(x)=0$;

$f^{'}(x)=-\ln{|x|}-1$, so $\space f^{'}(0^{+})=+\infty$.

2.

$f(x)=\frac{\arctan{x}}{\sqrt{|x|}}$.Verify:

$f(0^{+})=f(0^{-})=0$ (L'Hôpital's rule: $\lim_{x\to0^{+}}f(x)=lim_{x\to0^{+}}\frac{\frac{1}{1+x^2}}{\frac{1}{2\sqrt{x}}}=im_{x\to0^{+}}\frac{2\sqrt{x}}{1+x^2}=0$)

$f^{'}(x)=\frac{1}{(1+x^2)\sqrt{x}}-\frac{\arctan{x}}{2x\sqrt{x}}\to\infty\space$ as $x\to0^{+}$