Function approximation: why does using increasingly higher order approximating polynomials not lead to convergent approximation schemes in general?

Given some equally spaced data for a function on an interval, we can use line segments (pictured below), quadratic functions, various cubic schemes, and increasingly higher order polynomials to approximate the function using this equally spaced data.

For the linear scheme I show as an example in the figure above, as $h \rightarrow 0$, the approximation error also approaches zero, so we say the scheme is "convergent".

In general though, using increasingly higher order polynomials as approximations on the equally spaced intervals does not guarantee convergence. Wikipedia has an explanation that says that $h^{n+1}f^{(n+1)}(\xi)$ may not approach zero as $h$ approaches zero, if $f^{(n+1)}(\xi)$ dominates. Can you show this in a problematic case using pictures?