This is from Apostol's Calculus Vol 2
I have problem with $\lim\limits_{h\to0}\frac{||r(t+h)-r(t)||}{h}$: I think that limit may assume two different values depending on if we approach from left or right: $\frac{||r(t+h)-r(t)||}{h}=\sqrt{\frac{{(r_1(t+h)-r_1(t))^2+\dots+(r_n(t+h)-r_n(t))^2}}{h^2}}$ if $h>0$ and $\frac{||r(t+h)-r(t)||}{h}=-\sqrt{\frac{{(r_1(t+h)-r_1(t))^2+\dots+(r_n(t+h)-r_n(t))^2}}{h^2}}$ if $h<0$.
So how do we derive (8.15)?
The point is that $\left|\frac{||r(t+h)-r(t)||}{h}\right|=||\frac{r(t+h)-r(t)||}{h}||$ is bounded when $h\rightarrow 0$. Also we have that $E(a,y)\rightarrow 0$ when $h\rightarrow 0$. So the result follows.