Questions about hyper-plane tangent to a point on a surface

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 $0$ ?

-

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.