When does the remainder term in the taylor series go to zero?
Theorem: Let $f\in C^{N+1}([\alpha,\beta])$ and $x,x_0\in(\alpha,\beta)$. Then
$$f(x)=f(x_0)+f'(x_0)(x-x_0)+\frac{1}{2}f''(x_0)(x-x_0)^2+...+\frac{f^{(N)}(x_0)}{N!}(x-x_0)^N+\frac{(x-x_0)^{N+1}}{N!}\int_0^1 (1-t)^Nf^{(N+1)}(x_0+t(x-x_0))dt$$
Where the remainder term, $R_N$, is
$$R_N=\frac{(x-x_0)^{N+1}}{N!}\int_0^1 (1-t)^Nf^{(N+1)}(x_0+t(x-x_0))dt$$
I'm really not understanding this concept, especially because I'm struggling to comprehend what $R_N$ actually means.