Question

Can a tangent line be vertical?

Answer 1

Yes.

Explanation:

Perhaps the clearest instance is the tangent lines to the circle `x^2+y^2=r^2` at the points `(r,0)` and `(-r,0)`.

Restricting the `y` values to non-negative reals, gets us `y = sqrt(r^2-x^2)` whose graph is the upper semicircle. Again, the tangent lines at `(r,0)` and `(-r,0)` are vertical line.

Other examples include

the tangent line to `y=root(3)x` at `x=0`

and

the tangent to `y = root(3)(x^2) = x^(2/3)` at `x=0`.

In terms of the derivative, `x=a` is a vertical tangent line to the graph of `f(x)` if (?and only if ?) `a` is in the domain of `f` and `lim_(xrarra)abs(f'(x))=oo`
(If the domain of `f` only includes one side of `a`, then the limit is taken from the side in the domain.)