Providing an estimate for $|x|/(1+\sqrt{|x|})^{2}$

Question

Let $f:[-1,1] \rightarrow \mathbb{R}$, $x \mapsto |x|/(1+\sqrt{|x|})^{2}$. Can we find constants $c,a \in \mathbb{R}$, $c>0,0

Answer 1

In short: no.

When $x\to 0^+$, you have $$ f(x) = \frac{x}{(1+\sqrt{x})^2} = x\left(1-2\sqrt{x} + o(x)\right) = x+o(x) $$ i.e. $$ \frac{f(x)}{x} \xrightarrow[x\to0^+]{}1. $$ From this, there is no hope to show $\lvert f(x)\rvert \geq C \lvert x\rvert^\alpha$ for some $\alpha\in(0,1)$ and $C>0$ on a neighborhood of $0$, as this would imply, for $x>0$, $$ \frac{f(x)}{x}=\frac{f(x)}{x^\alpha}\cdot \frac{x^\alpha}{x} \geq \frac{C}{x^{1-\alpha}} \xrightarrow[x\to0^+]{}\infty $$