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.


Since we're doing geometry, let's think kinematically, too. Imagine a point moving along your curve, in the direction of increasing $x.$ Furthermore, let a ray project from the point, tangent to the curve there (you might think of your curve as a lane, and the point a car in the night; imagine the headlights beaming straight ahead as the tangent ray at any location on the curve).
Then as you approach the cuspoid from the left, the ray points down; at that point therefore, we can say it points down from the left. Now, let the point turn without leaving the point; then the tangent ray points up to the right -- at the same point! That happens nowhere else on the curve. Thus, we have two possible candidates for the tangent using this definition of tangent. If we require the tangent at a point to be unique, then there is no way to pick one of the two at the cuspoid without making further assumptions (which will be more or less arbitrary). The same thing happens for a point approaching the offending point from the right along the curve.
Thus, the curve has no unique tangent at that point. The reason is that you can think of the two branches of the curve as separate curves intersecting at a nonzero angle at that point. Suppose, for example, that we have two semicircular arcs instead, of the same radius and touching externally. For more clarity, take the positive parts of $(x+1)^2+y^2=1$ and $(x-1)^2+y^2=1.$ Then if we do not consider out tangents as directed, we can have a unique tangent at the origin in this case, since the two arcs are parallel at that place. However, if we think in terms of functions, there is still no derivative at that point, since the derivative measures the slope of the tangent line wrt increasing $x.$ However, if you imagine our point moving along this curve from left to right and crossing the origin, then the tangent ray points in opposite directions immediately before and after the point. Thus, there is no derivative there. However, purely geometrically, it is possible to speak of a tangent in this case.
I hope I have contributed to your understanding.
