The semigroup theory (as presented in Pazy's book) give us theorems that ensures existence of solutions for the abstract cauchy problem
$$\left\{\begin{align*} \frac{d}{dt}u(t)=A[u(t)];&&t\in[0,\infty)\\ u(0)=f&& \end{align*}\right.,\tag{1}$$ where $$\frac{d}{dt}u(t)=\lim_{h\to 0}\frac{u(t+h)-u(t)}{h}.$$
So, in some applications of semigroup theory to partial differential equations, we need to rewrite a given problem in the form $(1)$.
Here is an example that was taken of Goldstein's book:
Question: since each limit is taken in a different sense, why can we formally identify $\partial w/\partial t$ with $du/dt$? Could someone explain me why does it make sense?
Thanks.