Let $f:I\to \mathbb{R}$, $f$ is $C^{n+1}$ and $a\in I$. Then $$f(a+h)=f(a)+f'(a)h+\cdots+\frac{f^{(n)}(a+\theta_n h)}{n!}\cdot h^n,$$ where $\theta_n = \theta(h)$ with $0<\theta<1$.
Show that if $f^{n+1}(a)\neq0$, then $\lim_{h\to 0} \theta_n(h)=\frac{1}{n+1}$.
I know Taylor-formula with Lagrange remainder, but I don't have idea how to calculate this limit.