# Example of a tangent line intersecting with multiplicity three in a smooth point

In these notes it is said that a tangent line to a smooth point $$p$$ of a curve $$C$$ can be characterised as the unique line $$L$$ such that $$mult( L\cap C,q_0)\geq 2,$$ where mult means intersection multiplicity. I was trying to write down an example of such a point with intersection multiplicity precisely three but couldn't manage. For intersection multiplicity four, I took a parabola of degree four and the tangent at the origin.

Can someone provide an example?

The $$x$$ axis meets the cubic $$y=x^3$$ with multiplicity $$3$$ at the origin. You just have to overcome your prejudice and allow a tangent line to cross the curve.