Given a parametric function such as the following
$$ \left\{ \begin{array}{c} x = x(t) \\ y = y(t) \\ \end{array} \right. $$
Can $\lim_{\triangle t \to t_0} {\triangle y \over \triangle x}$ exist even if $x'(t_0)$ and $y'(t_0)$ doesn't? I confonted this in my analysis textbook and it assumed that when both$x'(t)$ and $y'(t)$ exist while $x'(t) \neq 0$:
$$\lim_{\triangle t \to t_0} {\triangle y \over \triangle x} = {lim_{\triangle t \to 0} {\frac{\triangle t}{x(t_0+\triangle t) -x(t_0)}}\frac{y(t_0+\triangle t) - y(t_0)}{\triangle t}} =\frac{\lim _{\Delta t \rightarrow 0} \frac{y\left(t_{0}+\Delta t\right)-y_{0}}{\Delta t}}{\lim _{\Delta t \rightarrow 0} \frac{x\left(t_{0}+\Delta t\right)-x_{0}}{\Delta t}} = \frac{y'(t_0)}{x'(t_0)}$$
However $\lim _{x \rightarrow x_{0}} \frac{f(x)}{g(x)}$ is not always the same as$\frac{\lim _{x \rightarrow x_{0}} f(x)}{\lim _{x \rightarrow x_{0} g(x)}}$
So potentially x(t) and y(t) can both be undifferentiable while $\frac{\Delta y}{\Delta x}$ exist, can you help me provide an example of this circumstances?
I was thinking about $x(t) = y(t) = t^{1/3}$, though both x(t) and y(t) are not differentiable at 0, ${y(t)\over x(t)}$ seems to be just a regular straight line and have an $\lim_{\triangle t \to 0} {\triangle y \over \triangle x}$ at 0. Is this a valide example? And if it is, would you mind provide another example that is not only a straight line? I still did not come up with any yet.
I think this question can also apply to higher dimensions. In other words, can the tangent line at a point exist in higher demensions even if one of the components are undifferentiable? Please give me some examples in higher dimensions if possible.
Based on the comments below. Such a function do exist, and I’d assume it exists in higher dimensions. So here is my immediately follow up questions. Given r(t) as a vector function in the form of <x(t),y(t),z(t)>. So $r’(t_0)$ is defined as $ \lim _{\Delta t \rightarrow 0} \frac{r\left(t_{0}+\Delta t\right)-r\left(t_{0}\right)}{\Delta t} = <x’(t_0), y’(t_0), z’(t_0)>$ Does that mean even a tangent line do exist at t0, the function itself is not differentiable? So does that mean we will loss something when parametrizing a function?