Mathematics

A vertical tangent means the slope is undefined—which happens when the denominator of $\frac{dy}{dx}$ blows up. Since $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$, you need the denominator to be zero while the numerator stays nonzero. So the key condition is $\frac{dx}{dt} = 0$ at $t = t_0$, but $\frac{dy}{dt} \neq 0$ at that same point—this ensures the curve is actually moving vertically, not just stalling out.

A vertical tangent means the slope is undefined—which happens when the denominator of $\frac{dy}{dx}$ blows up. Since $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$, you need the denominator to be zero while the numerator stays nonzero. So the key condition is $\frac{dx}{dt} = 0$ at $t = t_0$, but $\frac{dy}{dt} \neq 0$ at that same point—this ensures the curve is actually moving vertically, not just stalling out.