I'm afraid, everything seems to be "in the wrong side".
By which I mean the following. You can look at the tangent space of a variety $X$ at a point $p$ as the vector space of derivations at $p$. That is, the linear maps
$$ l : {\cal C}^{\infty}(X,\mathbb{R}) \longrightarrow \mathbb{R} $$
which, for every pair of functions $f,g: X \longrightarrow \mathbb{R}$ verify
$$ l(fg) = f(p)l(g) + g(p)l(f) \ . $$
Forgetting for a moment about this restriction, what you are doing in order to construct the tangent space $T_pX$ is
$$ X \mapsto {\cal C}^\infty(X,\mathbb{R}) \mapsto {\cal C}^\infty(X,\mathbb{R})^* \ , $$
where by $E^*$ I mean the dual of the vector space $E$.
Writting this in a more "categorical" way, it looks like
$$ X \mapsto \mathbf{Var}(X, \mathbb{R}) \mapsto \mathbf{Vct}(\mathbf{Var}(X, \mathbb{R}), \mathbb{R}) \ . $$
Here $\mathbf{Var}$ is a "convenient" category of "varieties" and $\mathbf{Vct}$ the category of vector spaces.
Now, if you look at what happens with limits and colimits when you perform such operations, you get:
$$ \mathbf{Vct}(\mathbf{Var}(\mathrm{colim}_i X_i, \mathbb{R}), \mathbb{R}) = \mathbf{Vct}(\mathrm{lim}_i \mathbf{Var}(X_i, \mathbb{R}), \mathbb{R}) \ . $$
And I have no idea about what happens next: this limit is "in the wrong side" of the hom set.
For limits the problem is worse, I think:
$$ \mathbf{Vct}(\mathbf{Var}(\mathrm{lim}_i X_i, \mathbb{R}), \mathbb{R}) \ . $$
The limit is already "in the wrong side" right from the start.