Andrew's comments showed me that in my first answer I was misunderstanding several aspects of his question. Since I am still not entirely sure that I am capturing the spirit of the problem, let be begin this answer by stating in my own words (in very dry mathematical terms) what I interpret the question(s) to be, so that he or someone else can correct me if necessary.
Let $mathcal{F}$ be the set of smooth functions $f colon mathbf{R}^2 to mathbf{R}$ whose restriction to the $x$-axis is constant. Let $mathcal{C}$ be the set of smooth functions $c colon (0,1) to mathbf{R}$ such that for every $f in mathcal{F}$,
the function $f circ c colon (0,1) to mathbf{R}$ extends to a smooth function $[0,1) to mathbf{R}$ (in the sense that all one-sided derivatives exist at the origin and equal the one-sided limits of the corresponding derivatives), and
$lim_{t to 0^+} f(c(t)) = f(0,0)$.
Question 1: If $c in mathcal{C}$, must $lim_{t to 0^+} c(t)$ exist? (Actually, Andrew is also asking more generally what one can say about $c$ if the limit does not exist.)
Question 2: Is there a rule (function) $mathcal{C} to mathcal{F}^r$ for some positive integer $r$, say taking $c$ to $(f_{1,c},ldots,f_{r,c})$, such that $f_{1,c}$ is independent of $c$, and $f_{2,c}$ depends only on $f_{1,c} circ c$, and $f_{3,c}$ depends only on $f_{1,c} circ c$ and $f_{2,c} circ c$, and so on, together with a rule (function) $R$ that takes as input a sequence of smooth functions $(g_1,ldots,g_r)$ from $(0,1)$ to $mathbf{R}$ and outputs a point in $mathbf{R}^2$ such that for every $c in mathcal{C}$, we have $R(f_{1,c}circ c,ldots,f_{r,c} circ c) = lim_{t to 0^+} c(t)$ whenever the latter exists?
Question 3: Same as Question 2, but with $lim_{t to 0^+} c'(t)$ in place of $lim_{t to 0^+} c(t)$.
Answer to Question 1: No. A negative answer was essentially given already by Andrew himself. Namely, let $c(t) := (sin 1/t,e^{-1/t})$. This $c(t)$ has the following properties: the $x$-coordinate is bounded, the derivatives of the $x$-coordinate grow at most polynomially in $1/t$ as $t to 0^+$, and the $y$-coordinate and derivatives of the $y$-coordinate decay to $0$ exponentially as $t to 0^+$. As explained by Andrew, for any $f in mathcal{F}$, the chain rule shows that $f(c(t))$ extends to a smooth function $[0,1) to mathbf{R}$ whose value at $0$ is $f(0,0)$ and whose higher derivatives at $0$ are all $0$. $square$
Answer to Questions 2 and 3: Yes. In fact, we can do it with $r=2$, and with both $f_{1,c}$ and $f_{2,c}$ independent of $c$. Namely, use $f_1(x,y)=y$ and $f_2(x,y)=e^x y$. From $f_1 circ c$ and $f_2 circ c$, we may recover not only the $y$-coordinate of $c$ as $f_1 circ c$, but also the $x$-coordinate of $c$ as $log (f_2(c(t))/f_1(c(t)))$. So the whole function $c(t)$, and hence any property of $c(t)$, can be detected. $square$
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