My questions arise from 11.21 DEFINITON and 11.22 THEOREM ,both of them presented on p341 An Introduction To Analysis W.R.Wade 3ed.
Question 1.
Considering 11.21 DEFINITON, Let $S:=\begin{cases} x^{2}+y^{2}=1 \\ z=1 \end{cases}$ is a curve in $\mathbf{R}^3.\mathbf{c}=(1,0,1)\in S.$ For every $\mathbf{n}=(x,0,z),x,z\in\mathbf{R}$ satisfy $\mathbf{n}\cdot\frac{\mathbf{c}_k-\mathbf{c}}{||\mathbf{c}_k-\mathbf{c}||}\rightarrow 0$ for all sequences $\mathbf{c}_k\in S\backslash \begin{Bmatrix} \mathbf{c} \end{Bmatrix} $ that converge to $\mathbf{c}.$ Then we leads to planes being tanget to curve ,also we have so much different planes $\Pi_{\mathbf{n} }$ with normal $\mathbf{n}=(x,0,z)$ to be tangent to $S$ at $\mathbf{c}.$ Those aren't quite the same as our's common sense.So what is the formal definition of tangent hyperplanes?
Question 2.
Considering 11.22 THEOREM,If we find another $\mathbf{p}$ satisfied $\mathbf{p}\cdot\frac{\mathbf{c}_k-(\mathbf{a},f(\mathbf{a}))}{||\mathbf{c}_k-(\mathbf{a},f(\mathbf{a}))||}\rightarrow 0$ for all sequences $\mathbf{c}_k\in S\backslash \begin{Bmatrix} (\mathbf{a},f(\mathbf{a})) \end{Bmatrix} $ that converge to $(\mathbf{a},f(\mathbf{a})),$ How can I prove there must be a $ \lambda\in \mathbf{R},$such that $\mathbf{p}=\lambda\cdot \mathbf{n},$where $\mathbf{n}=(\nabla f( \mathbf{a}),-1)?$ That is to prove there is an unique hyperplane to be tangent to $S$ at $(\mathbf{a},f(\mathbf{a})).$ I try to make use of the uniqueness of $\nabla f(\mathbf{a})$ to get the uniqueness of hyperplane at $(\mathbf{a},f(\mathbf{a})),$ but I haven't got the answer yet.
Any of your help will be appreciated!