Parametric motion question What exactly happens when both $\frac{\mathrm{d}y}{\mathrm{d}t}$ and $\frac{\mathrm{d}x}{\mathrm{d}t}$ equal zero?
I know that if $\frac{\mathrm{d}y}{\mathrm{d}t} =0$ then its a vertical tangent with $\frac{\mathrm{d}x}{\mathrm{d}t} \neq 0$
and if $\frac{\mathrm{d}x}{\mathrm{d}t}=0$ then its a horizontal tangent if $\frac{\mathrm{d}y}{\mathrm{d}t}\neq 0$.  
Do you apply L'Hoptial rule to see if it is a tangent? 

$$\dfrac{dy}{dx} = \dfrac{\frac {dy}{dt}}{\frac{dx}{dt}}$$
In your case, that means $\dfrac{dy}{dx}$ is of indeterminant form.
If both derivatives are zero, the tangent vector does not have a well-defined direction; in physical terms this indicates that the moving point has "slowed down" to a halt at the time $t$. 
In this case, I would express the parametrization as a function of $x$ by solving for $t$ in the parametrization of $x(t)$, and substituting $t(x)$ into the parametrization of $y$ to get an equation in terms of $x$ and $y$, then differentiate to find $dy/dx$ directly.
