Perhaps the clearest instance is the tangent lines to the circle x^2+y^2=r^2x2+y2=r2 at the points (r,0)(r,0) and (-r,0)(−r,0).
Restricting the yy values to non-negative reals, gets us y = sqrt(r^2-x^2)y=√r2−x2 whose graph is the upper semicircle. Again, the tangent lines at (r,0)(r,0) and (-r,0)(−r,0) are vertical line.
Other examples include
the tangent line to y=root(3)xy=3√x at x=0x=0
and
the tangent to y = root(3)(x^2) = x^(2/3)y=3√x2=x23 at x=0x=0.
In terms of the derivative, x=ax=a is a vertical tangent line to the graph of f(x)f(x) if (?and only if ?) aa is in the domain of ff 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.)
