I apologize if the answer to my question is already on the Internet somewhere, but if it is, I couldn't find it, nor can I prove it myself.
The question is a short one: given a suitably "nice" parametric function $$ T(t) = (x(t), y(t)) $$ if we have the asymptotic relationships $$ \begin{align*} x(t) &\sim w(t) &(\text{as } t \rightarrow t_0)\\ y(t) &\sim z(t) &(\text{as } t \rightarrow t_0)\\ \end{align*} $$ and we define $$ U(t) = (w(t),z(t)) $$ then does this imply that $$ T(t) \sim U(t) \ (\text{as } t \rightarrow t_0) $$ where $t_0$ $\in \mathbb{R}\ \cup \ \{-\infty, \infty\}$? I imagine so, as the $x$ and $y$ components are both asymptotic, so it would seem that the overall functions should be asymptotic too, but I don't know how to prove such a thing. For all functions I have tested this relation is true, but I don't know how I might go about proving such a thing. Furthermore, if there is a proof for this relation, is there a good criteria for where this type of relation might fail?