【答案】B

【解析】①\(\lim\limits_{x \to 0}\frac{|f(x)| - f(0)}{x} = A\).

\(\Rightarrow |f(0)| - f(0) = 0 \Rightarrow f(0) = |f(0)| \Rightarrow f(0) = 0\). 或\(f(0)\)为正.

若\(f(0)\)为正，则当\(x \to 0\)时. 有 \(f(x) > 0\).则\(\lim\limits_{x \to 0}\frac{|f(x) - f(0)|}{x} = \lim\limits_{x \to 0}\frac{f(x) - f(0)}{x} = A \Rightarrow f'(0) = A\)

若\(f(0) = 0\)，由\(\lim\limits_{x \to 0}\frac{|f(x)| - f(0)}{x} = \lim\limits_{x \to 0}\frac{|f(x)|}{x} = A\)

\(\Rightarrow \begin{cases}\lim\limits_{x \to 0^+}\left|\frac{f(x)}{x}\right| = A \\ \lim\limits_{x \to 0^-}\left|-\frac{f(x)}{x}\right| = A \end{cases} \Rightarrow\)若\(A = 0\)，则\(f'(0)\)存在，若\(A \neq 0\)，则\(f'(0)\)不存在.

②由\(\lim\limits_{x \to 0}\frac{f(x) - |f(0)|}{x} = A \Rightarrow f(0) = |f(0)|\) ，则\(f(0) = 0\)或\(f(0) > 0\)

若\(f(0) = 0\)，则\(A = \lim\limits_{x \to 0}\frac{f(x) - |f(0)|}{x} = \lim\limits_{x \to 0}\frac{f(x)}{x} = f'(0)\)

若\(f(0) > 0\)，则\(A = \lim\limits_{x \to 0}\frac{f(x) - f(0)}{x} = f'(0)\) 从而②成立.

③由\(\lim\limits_{x \to 0}\frac{|f(x)|}{x}\)存在，则 \(|f(0)| = 0 \Rightarrow f(0) = 0\).  则同①，从而③错.

④ \(\lim\limits_{x \to 0}\frac{|f(x)| - |f(0)|}{x}\)存在，则\(|f(x)|\)在\(x = 0\)处可导\(\Rightarrow f(x)\)在\(x = 0\)处可导，④正确.

从而①③错误，②④正确，选B.