Let $A$ be a ring (say finitely generated algebra over an algebraically closed field). Then, does $\varinjlim \mathrm{Spec} A[t]/(t^m)$ exist (in the category of schemes)? And if it does, then is it equal to $\mathrm{Spec} A[[t]]$?

Edit based on comment below: The result holds easily when $A=k$ is a field. Then, $\mathrm{Spec} k[t]/(t^n)$ only has one point and hence, for any scheme $X$ such that we have a family of morphisms $\mathrm{Spec} k[t]/(t^n) \to X$, we can assume that $X$ is affine and the result follows from the equivalence of the category $Ring^{op}$ and the category of affine schemes.