I'm reading the book Physics, by Tipler, and I'm confused at the following statement:
[...] From the first principle of thermodynamics, $\Delta U = Q+ W$. Suppose an ideal gas is given heat while keeping it's volume constant, then $W=0$ and $Q=Q_v:= C_V\Delta T$, where $C_V$ is the heat capacity of the gas, then $$ \Delta U = C_V\, \Delta T $$ Taking the limit as $\Delta T$ goes to $0$, we get $$ d\,U= C_V \, d\, T $$
I don't understand that last "equation", taking the limit we should just get $\lim_{\Delta T} U = 0$, and nothing more. I've taken calculus, introductory real analysis, and a few other math courses (I'm a math student) and I still don't understand these famous differentials.
Does the statement in the book make any sense?