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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?

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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.

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