Can a tangent line be vertical?

1 Answer
Jan 29, 2016

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.)