As I'm working from do Carmo's Differential Geometry of Curves and Surfaces, I have found some of his imprecise language regarding strong and weak tangents to be most irksome. I've seen similar posts floating around this site from those suffering the same woe. See my comments here.
In do Carmo's words:
Let $\alpha : I \to \mathbb{R}^3$ be a simple curve of class $C^0$ (i.e. continuous) ... We say that $\alpha$ has a strong tangent at $t=t_0$ if the line determined by $\alpha(t_0+h)$ and $\alpha(t_0+k)$ has a limit position when $h,k \to 0$.
(Note: we'd obviously like to readily generalize to $\mathbb{R}^n$.)
Nowhere in the text does do Carmo actually define a "limit position," but it seems easy to deduce what he's going for.
Is the following a valid, rigorous definition of a strong tangent?
We say that $\alpha$ has a strong tangent at $t=t_0$ if the limit $$\lim_{(h,k) \to \vec{0}} \dfrac{[\alpha(t_0+h)-\alpha(t_0+k)]/(h-k)}{\left\lvert [\alpha(t_0+h)-\alpha(t_0+k)]/(h-k) \right\rvert}$$ exists and is not equal to the zero vector (such that the limit determines a line).
I came to that definition after playing around with limits in Mathematica for a good while. If that doesn't work, then what does? Is the normalization inside the limit necessary?
A motivating example is the curve $\alpha(t)=(t^3,t^2)$ with $t \in \mathbb{R}$, which has a weak tangent at $t=0$, but not a strong tangent. It's easy to examine the slope $[y(t_0+h)-y(t_0+k)]/[x(t_0+h)-x(t_0+k)]$, but that method doesn't easily generalize to $\mathbb{R}^n$.