What is the tangent at a sharp point on a curve? How to know which line represents tangent to a curve $y=f(x)$ (in RED) ?From the diagram , I cannot decide which line to take as tangent , all seem to touch at a single point.


If the cuspid you draw is actually a cuspid (roughly speaking: if microscopically it is not "rounded") you can speak of a right and left tangent, same as you do with limits.   
Refer to this wiki article.
Also note that, if the semi-derivative on one side has the value of $m$ and the other $-\,m$, then the two lines geometrically coincide into a single line (or, maybe better, half line or ray).
