Kinematic formulas for constant $n$th derivative of position

I was wondering how to solve for $x(t)$ in the general case of constant $n$th derivative of $x$. This means to solve the equation $$frac{mathrm{d}^n x}{mathrm{d} t^n}=q,$$ where $q$ is a constant. Looking at the formulas for constant acceleration and snap, I guess that the solution is $$x(t)=sum^n_{k=0} frac{q_{(k),0}t^k}{k!}$$, where $q_{(k),0}$ is the initial value of the $k$th derivative of $x$.

My question is now, how to rigorously derive this result if it is correct? What about the limit as $n rightarrow infty$? It seems that it tends to an exponential function. Why this physically happens?