Is there a geometrical definition of a tangent line?

A tangent line may or may not cross the curve at the point of tangency, but among all lines through the point of tangency it is always at the boundary between those that cross the curve in one direction at that point and those that cross it in the other direction at that point.

I believe there is no good geometric definition of a tangent line: at least, no definition that covers all cases. We need calculus for a good definition. Here is a quote from the book Calculus: Graphical, Numerical, Algebraic by Ross L. Finney et al., page 84.

The problem of how to find a tangent to a curve became the dominant mathematical problem of the early seventeenth century and it is hard to overestimate how badly the scientists of the day wanted to know the answer. Descartes went so far as to say that the problem was the most useful and most general problem not only that he knew but that he had any desire to know.

The textbook then gives the usual calculus definition using limits.

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To see the difficulty of finding a definition that always works, try using geometry to explain why the $x$-axis is the tangent line to the curve

$$f(x) = \left\{ {\begin{array}{*{20}{c}} {{x^2}\sin \frac{1}{x},}&{x \ne 0} \\ 0,&{x = 0} \end{array}} \right.$$

at $x=0$.

Excluding inflection points, we could state, a tangent line through a given point on a curve is the boundary of a half-plane which contains some segment of the curve to either side of the point, and for which the point itself is on the boundary.