Is there a specific example when the analytic form of a derivative $\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$ is preferred to the numerical form $\frac{f(x+h)-f(x)}{h}$, $h \ll 1$? Are there cases when the numerical approach fails?
I'm trying to figure out why in the secondary school we focus on the analytic form and not on the numerical one. In my opinion, the numerical form is much easier to understand since it doesn't require a difficult mental transition from a secant to a tangent. In addition, the numerical approach provides good accuracy for calculating the derivative. Finally, calculators and computers use the numerical approach to plot derivatives so that they can be easily visualized.
On the other hand, we can say that the analytic form is important for definition of continuity, Taylor series etc. These topics, however, are higher level mathematics and can be left for the classes with advanced math or even college.