Vertical tangents of parametric curves

[ratios]

I learned that vertifcal tangents occur at a parametric curve if the derivative of the curve is undefined. That is given dy/dx = dy/dt / dx/dt, a vertical tangent occurs when dx/dt = 0.

I don't understand why this is so. I know that vertical tangents occur when the slope is infinite, but if dx/dt = 0, dy/dx is undefined. I understand that when the slope is infinite it is undefined, but there are cases in which the slope is undefined, and yet it is not infinite, right? Or is it true that for parametric curves, all undefined slopes are infinite?

 

[HallsofIvy]

You might want to recheck your information. For one thing, I'm not clear what you mean by "the derivative of the curve"!

In any case, the curve given by x= t², y= t² has dy/dt= 2t, dx/dt= 2t which are 0 at t= 0 and so dy/dx is "undefined" (according to you). But obviously that curve is just y= x which has derivative 1 everywhere and never has a vertical tangent.

My reason for writing "(according to you)" was that many books use the term "undefined" only for the case when the denominator is 0 and the numerator is NOT and use the term "undetermined" for 0/0. "Undefined" in that sense (here, dx/dt= 0 but dy/dt does not) does in fact mean there is a vertical tangent.