the slope of the line $C$ is the derivative of $f$ at $x_0$, $f'(x_0)$, defined by $$f'(x_0)=\lim_{\Delta x\to 0}\frac{f(x_0+\Delta x)-f(x_0)}{\Delta x}.$$ More informally again, this limit means that you can get "arbitrarily close" to $f'(x_0)$ by taking $\Delta x$ "sufficiently small" (but not $0$) in the difference quotient $\frac{f(x_0+\Delta x)-f(x_0)}{\Delta x}$,