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If a one-variable function is sufficiently smooth, then the limit position ofsecant lines its graph is a tangent line. By analogy, one would expect that thelimit position of secant planes of a two-variable smooth function is a planetangent to its graph. Amazingly, this is not necessarily true, even when thefunction is a simple polynomial. Despite this paradox, we show that someanalogies with the one-variable case still hold in the multi-variable context,provided we use a particular vector product: the Clifford's geometric one.

The surface tangent paradox and the difference vector quotient of a secant plane