Tangent line to a differentiable curve

Question

Let $\alpha: I \to \mathbb{R}^{n}$ be a differentiable curve such that $\alpha'(a) \neq 0$ for some $a \in I$. The line $L \subset \mathbb{R}^{n}$ through $\alpha(a)$ is the tangent line to the curve $\alpha$ at that point if and only if $$ \lim_{t \to a} \frac{d(\alpha(t), L)}{|\alpha(t) - \alpha(a)|} = 0, $$ where $d(\alpha(t), L) = \inf\{|\alpha(t) - \alpha(a)|;\,\alpha(a) \in L\}$ denotes the distance between the point $\alpha(t)$ and the line $L$.

I really need some help with this exercise. Maybe a hint.

Answer 1

A picture would help. I think it's asking whether $\lim\sin\theta\to0$, of the angle between the tangent and the chord to the approaching point on the curve. Well, that chord aligns with the tangent in the limit.