When the derivative does not exist at a point , what are the possible graphs at that point?

Question

Can there be a vertical asymptote at that point? Can there be a hole at that point? Can there be discontinuity at that point? Which one's are yes and which one's are no? (True, False). Thanks.

Answer 1

Hi Will, 

Remember the derivative of a function f(x) is defined as 

 

             l            (f(x+h)-f(x))

     f'(x) =lim      -----------------

              h-->0          h

If the derivative does not exist that means the limit does not exist.

Yes, there can be a vertical asymptote at the point where the derivative does not exist because the limit from the left-hand side and the limit from the right-hand side do not approach the same value at the location of the vertical asymptote. The limits diverge in opposite directions.

Yes, the function is discontinuous at the point where the derivative does not exist, because the slopes of the tangent lines or the derivatives are different on either side of the point of discontinuity.