设函数 $y=f(x)$ 在点 $x=0$ 处连续,则下列命题中,错误的是( )
A. 若 $\lim _{x \rightarrow 0} \frac{f(x)}{x}=a$, 则 $f^{\prime}(0)=a$.
B. 若 $f(0)=0, f^{\prime}(0)=a$, 则 $\lim _{x \rightarrow 0} \frac{f(x)}{x}=a$.
C. 若 $\lim _{x \rightarrow 0} \frac{f(x)}{x^2}=\frac{a}{2}$, 则 $f^{\prime \prime}(0)=a$.
D. 若 $f(0)=f^{\prime}(0)=0, f^{\prime \prime}(0)=a$, 则 $\lim _{x \rightarrow 0} \frac{f(x)}{x^2}=\frac{a}{2}$.