I am stuck on trying to prove that the approximate tangent space of a submanifold of $\mathbb{R}^n$ agrees with its tangent space. To make things more precise I'll give the relevant definitions. Note that $\mathcal{H}^k$ denotes the $k$-dimensional Hausdorff measure on $\mathbb{R}^n$ and $\mathcal{L}_n$ denotes Lebesgue measure on $\mathbb{R}^n$.
Definition. (Approximate tangent space) Let $E\subseteq\mathbb{R}^n$ be a $\mathcal{H}^k$-measurable set such that $\mathcal{H}^k(E\cap K)<\infty$ for all compact $K\subseteq\mathbb{R}^n$. and let $P$ be a $k$-dimensional subspace of $\mathbb{R}^n$. We say that $P$ is the approximate tangent space of $E$ at $x\in E$ if $$\lim_{\lambda\downarrow0}\int_{\frac{E-x}{\lambda}}f~\mathrm{d}\mathcal{H}^k=\int_P f~\mathrm{d}\mathcal{H}^k$$ for all compactly supported continuous $f:\mathbb{R}^n\to\mathbb{R}$.
Definition. (Tangent space) Let $M$ be a $k$-dimensional $C^1$-submanifold of $\mathbb{R}^n$, and let $x\in M$. We define the tangent space of $M$ at $x$, denoted $\mathrm{T}_xM$, as the subspace of $\mathbb{R}^n$ consisting of all vectors $v$ such that $v=\gamma'(0)$ for some $C^1$-curve $\gamma:(-1,1)\to M$ with $\gamma(0)=x$.
What I am trying to prove is that if $M$ is a $k$-dimensional $C^1$-submanifold of $\mathbb{R}^n$, then $\mathrm{T}_xM$ is also the approximate tangent space of $M$ at $x$. In particular then what I need to show is that
$$\lim_{\lambda\downarrow0}\int_{\frac{M-x}{\lambda}}f~\mathrm{d}\mathcal{H}^k=\int_{\mathrm{T}_xM} f~\mathrm{d}\mathcal{H}^k$$
for all compactly supported continuous $f:\mathbb{R}^n\to\mathbb{R}$. Before giving my attempt, I'll also state the Area formula, which I'm using.
Theorem. (Area formula) Let $f:\mathbb{R}^n\to\mathbb{R}^m$ be Lipschitz, where $n\leq m$, and let $g\in\mathscr{L}^1(\mathbb{R}^n)$. Then $$\int_{\mathbb{R}^n}gJf~\mathrm{d}\mathcal{L}_n=\int_{\mathbb{R}^m}\sum_{x\in f^{-1}(\{y\})}g(x)~\mathrm{d}\mathcal{H}^n(y),$$ where $$Jf(x)=\sqrt{\det(Df(x)^*\circ Df(x))},$$ with $^*$ denoting the adjoint.
The following is what I've tried so far:
We consider first when we can parameterize all of $M$ by a $C^1$-map $\psi:V\to M$ on some open set $V\subseteq\mathbb{R}^k$. Suppose without loss of generality that $\psi(0)=x$. We have that
$$\mathrm{T}_xM=\mathrm{span}\{\partial_1\psi(0),\dots,\partial_k\psi(0)\},$$
and so in particular we can consider the function $h:\mathbb{R}^k\to\mathrm{T}_xM$ defined by
$$h(y)=D\psi(0)y,$$
which gives us an isomorphism, and in particular $h$ is Lipschitz with
$$Jh(y)=J\psi(0)$$
for all $y\in\mathbb{R}^k$, and so by the Area formula we have that
$$\int_{\mathrm{T}_xM} f~\mathrm{d}\mathcal{H}^k=\int_{\mathbb{R}^k}f(D\psi(0)y)J\psi(0)~\mathrm{d}\mathcal{L}_k(y).$$
Now take $\lambda>0$, define $\eta_\lambda:\mathbb{R}^n\to\mathbb{R}^n$ by
$$\eta_\lambda(y)=\frac{y-x}{\lambda},$$
and observe that $\eta_\lambda\circ\psi$ then parameterizes $\frac{M-x}{\lambda}$, and
$$J(\eta_\lambda\circ\psi)=\frac{1}{\lambda^k}J\psi.$$
We thus have, again by the Area formula, that
$$\int_{\frac{M-x}{\lambda}} f~\mathrm{d}\mathcal{H}^k=\int_V f\left(\frac{\psi(y)-x}{\lambda}\right)\frac{J\psi(y)}{\lambda^k}~\mathrm{d}\mathcal{L}_k(y).$$
At this point I suppose I want to let $\lambda\downarrow0$ and use some convergence theorem to show that this integral converges to the previous one. I am however stuck on how to actually do this, and so need help on how to progress from here (assuming my idea even works and is correct so far).
Edit: So I realized that if $y\neq0$, then $\psi(y)\neq x$, which means that $\frac{\psi(y)-x}{\lambda}$ is outside of the support of $f$ for sufficiently small $\lambda$. This means that we have that
$$\lim_{\lambda\downarrow0}f\left(\frac{\psi(y)-x}{\lambda}\right)\frac{J\psi(y)}{\lambda^k}=0$$
for all $y\neq0$. I'm a bit unsure about how useful this actually is right now, but I'll try to investigate what happens when $y=0$ when I get home, and hopefully get some useful behavior out of the integral.
Alright so I eventually came up with a solution which I think works. So observe that, by what was added in the edit, we can, for each $\varepsilon>0$ such that $B_\varepsilon(0)\subseteq V$, write
$$\int_Vf\left(\frac{\psi(y)-x}{\lambda}\right)\frac{J\psi(y)}{\lambda^k}~\mathrm{d}\mathcal{L}_k(y)=\int_{B_\varepsilon(0)}f\left(\frac{\psi(y)-x}{\lambda}\right)\frac{J\psi(y)}{\lambda^k}~\mathrm{d}\mathcal{L}_k(y)$$
for all sufficiently small $\lambda$. Now performing the change of variables $\xi=\frac{y}{\lambda}$ we have that
$$\int_{B_\varepsilon(0)}f\left(\frac{\psi(y)-x}{\lambda}\right)\frac{J\psi(y)}{\lambda^k}~\mathrm{d}\mathcal{L}_k(y)=\int_{B_{\frac{\varepsilon}{\lambda}}(0)}f\left(\frac{\psi(\lambda\xi)-x}{\lambda}\right)J\psi(\lambda\xi)~\mathrm{d}\mathcal{L}_k(\xi).$$
Consider now the expansion
$$\frac{\psi(\lambda\xi)-x}{\lambda}=D\psi(0)\xi+o(\lVert \lambda\xi\rVert)$$
as $\lambda\xi\to0$. This gives us that
$$\lim_{\lambda\downarrow0}f\left(\frac{\psi(\lambda\xi)-x}{\lambda}\right)J\psi(\lambda\xi)=\lim_{\lambda\downarrow0}f\left(D\psi(0)\xi+o(\lVert \lambda\xi\rVert)\right)J\psi(\lambda\xi)=f(D\psi(0)\xi)J\psi(0).$$
We can thus apply the DCT to get that
$$\lim_{\lambda\downarrow0}\int_{B_{\frac{\varepsilon}{\lambda}}(0)}f\left(\frac{\psi(\lambda\xi)-x}{\lambda}\right)J\psi(\lambda\xi)~\mathrm{d}\mathcal{L}_k(\xi)=\int_{\mathbb{R}^k}f(D\psi(0)\xi)J\psi(0)~\mathrm{d}\mathcal{L}_k(\xi),$$
from which the result now follows.