I know that \cos'(0) is $0$, but my work follows:
\begin{align}\cos'(0) &= \lim_{\Delta x\to 0}\frac{f(0+\Delta x) - f(0)}{\Delta x}\\ &= \frac{\cos(0+\Delta x) - \cos(0)}{\Delta x}\\ &= \frac{\cos(0) - \cos(0)}{0} = \frac{0}{0}\end{align}
Where have I gone wrong? What can I do to show that it equals 0?
(Note: No derivative rules are allowed in my calculus class yet, just difference quotients)