calculus - Limits- h'(0) and writing properly

1. Write the definition of the derivative of a function of $x$ as a $\lim_\limits{\Delta x\to0}$

I have no idea what this is talking about.

2. Consider
   $H(x) =
   \begin{cases}
   x(1-cos(\frac{1}{x})) & \text{x$\neq$0} \\
   0, & \text{$x=0$}
   \end{cases}$
   Using the limit definition of the derivative find $H'(0)$. If that limit does not exist, explain why.
   I feel like the limit does exist, and I'm just missing something.
   
   A reply today would be appreciated.