Searching for a sequence of functions Consider the following set of functions:
$$
A=\left\{f\in C([0,1],\mathbb{R}): f(0)=0, \lim_{r\searrow 0}\frac{f(r)}{r}\text{ exists}\right\}.
$$ 
Is there a sequence $(f_n)\in A^{\mathbb{N}}$ such that $f_n\searrow 0$ but 
$$
\lim_{n\to\infty}\lim_{r\searrow 0}\frac{f_n(r)}{r}\neq 0?
$$
Maybe you can give me some hints since I did not find such a sequence yet.

$f_n(x) = x (1-x)^n$ is an example.
