I have a doubt about exercise 1.3.7 from the book Differential Geometry of Curves and Surfaces by Do Carmo.
The exercise asks to prove that a curve $\alpha$ which is $C^1$ at $t_0$ and $\alpha'(t_0)\neq 0$ has a strong tangent at $t_0$. The definition of strong tangent can be found here.
My attempt: The line that goes trough $\alpha(t_0+h)$ and $\alpha(t_0+k)$ is $$\{ \lambda(\alpha(t_0+h)-\alpha(t_0+k))+\alpha(t_0+h):\lambda\in \mathbb{R} \}.$$
I don't understand how the limit $h,k\to 0$ should be taken in the above expression. As $h,k\to 0$, the line should become $$\{ \lambda \alpha'(t_0)+\alpha(t_0):\lambda\in \mathbb{R} \}.$$
Do Carmo gives the following hint to solve this exercise $$\frac{\alpha(t_0+h)-\alpha(t_0+k)}{h-k}\xrightarrow{h,k\to0}\alpha'(t_0).$$ I understand why this is true (Mean Value Theorem) but I don't know what this tells us about the line.
Thanks!