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I am quite confused with the tangent of a hyperplane to a point. My question is why it consider $\lim_{x\to x_0} |x-x_0|^{-1}||z'-z|=0$instead of just $\lim_{x\to x_0}|z'-z|=0$ and why would it be $0$ ?

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I can't make any sense of the expression "the tangent of a hyperplane to a point"; I'll assume that it's meant to refer to what you correctly refer to as the "hyperplane tangent to a point on a surface" in the title.

Your version of the condition would be satisfied whenever the plane intersects the surface at $\mathbf x$. A tangent plane is meant to do more than just intersect. The English word tangent is derived from the present participle tangens of the Latin verb tangere, to touch. The tangent touches the surface in the sense that it coincides with the surface not only at $\mathbf x$ but also to first order in its vicinity. This is what's expressed by the condition in the text.

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