Question: Show that $\alpha(t)=(t^3,t^2)$, $t\in \Bbb R$, has a weak tangent but not a strong tangent at $t=0$.
(Weak tangent) $\alpha: I \to \Bbb R^3$ has a weak tangent at $t_0 \in I$, if the line determined by $\alpha(t_0 + h)$ and $\alpha(t_0)$ has a limit position when $h \to 0$.
(Strong tangent) $\alpha: I \to \Bbb R^3$ has a strong tangent at $t_0 \in I$, if the line determined by $\alpha(t_0 + h)$ and $\alpha(t_0 + k)$ has a limit position when $h \to 0$ and $k \to 0$.
My query:
I'm not really clear what argument to use to demonstrate this. The weak tangent is the line joining $\alpha(t_0)$ and $\alpha(t_0+h)$, which is $$ (\lambda(x(t_0+h)-x(t_0))+x(t_0), \lambda(y(t_0+h)-y(t_0))+y(t_0)) $$ If $t_0=0$ then $x(t_0)=0$ so this becomes $$ (\lambda h^3, \lambda h^2) $$ The strong tangent is $$ (\lambda(x(t_0+h)-x(t_0+k))+x(t_0+k), \lambda(y(t_0+h)-y(t_0+k))+y(t_0+k)) $$ $$ =(\lambda (h^3-k^3)+k^3,\lambda (h^2-k^2)+k^2) $$ As $h,k\rightarrow0$ this seems badly defined. But how can I make this argument precise?
Also, what is the intuitive meaning of the strong and weak tangents?
[This is exercise 1-3-7 of Differential Geometry of Curves and Surfaces by Do Carmo.]