Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a function. We say that $f$ has
$$\text{PROPERTY 1 if } \lim_{h\rightarrow 0} \frac{f(h) - f(0)}{\sqrt{|h|}} \text{ exists and is finite, and}$$
PROPERTY 2 if $\lim_{h\rightarrow 0} \frac{f(h) - f(0)}{h^2}$ exists and is finite.
Then which of the following options is/are correct?
(A) $f(x) = |x|$ has PROPERTY 1
(B) $f(x) = x^{2/3}$ has PROPERTY 1
(C) $f(x) = x|x|$ has PROPERTY 2
(D) $f(x) = \sin x$ has PROPERTY 2