When Can I Use That Rule?

Calculus Level 2

limx0sinx1cosx=limx0ddx(sinx)ddx(1cosx)(1)=limx0cosxsinx(2)=limx0ddx(cosx)ddx(sinx)(3)=limx0sinxcosx(4)=sin0cos0=0(5) \begin{array} {l c l r } \displaystyle \lim_{x\to0} \dfrac{\sin x}{1-\cos x} &= &\displaystyle \lim_{x\to0} \dfrac{\frac d{dx}(\sin x)}{\frac d{dx} (1-\cos x)} & \quad (1) \\ \displaystyle &= &\displaystyle \lim_{x\to0} \dfrac{\cos x}{\sin x} & \quad (2) \\ \displaystyle &= &\displaystyle \lim_{x\to0} \dfrac{\frac d{dx}(\cos x)}{\frac d{dx}(\sin x)} & \quad (3) \\ \displaystyle &= &\displaystyle \lim_{x\to0} \dfrac{-\sin x}{\cos x} & \quad (4) \\ \displaystyle &= & -\dfrac{\sin0}{\cos 0} = 0 & \quad (5) \end{array}

The above is my attempt at proving that limx0sinx1cosx=0 \displaystyle \lim_{x\to0} \dfrac{\sin x}{1-\cos x} = 0 . In which step did I first commit a flaw in my logic?

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