Parametric equations: $\ \frac{dy}{dt}=0\ \nRightarrow\ $ the tangent is parallel to the x-axis?

Question

Suppose we have real parametric equations defined by:

$y = f(t),\quad x = g(t).$

The book I am using says that:

At a point where the tangent is parallel to the x-axis, $\large{\frac{dy}{dt}}=0.$

At a point where the tangent is parallel to the y-axis, $\large{\frac{dx}{dt}}=0.$

This makes sense to me. But then the book says that the converse is not true:

For example, $\ \frac{dy}{dt}=0\ \nRightarrow\ $ the tangent is parallel to the x-axis.

This says that, loosely speaking, for a function where a small change in $t$ causes almost no change in $y$, the tangent is not necessarily parallel to the x-axis.

But I can't think of a counterexample. Is there an example where the implication does not hold?

Answer 1

Imagine a system like $x=\sin(t)$ and $y=\sin(t)$. $\frac{dy}{dt}=0$ when $t=\frac\pi2$, but the graph of the system has a slope of 1 everywhere that the slope is defined.

Answer 2

$$\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} $$ The only way for this to not equal zero (when $\frac{dy}{dt}=0$) is when $\frac{dx}{dt} $ is also zero, in which a limit of the form $\frac 00$ must be taken, which can evaluate to any real number.