Farlow book PDEs for Scientists and Engineers pg. 27 shows derivation for Heat Equation. It starts by stating
Net change of heat inside $[x,x+\Delta x]$ = Net flux of heat across boundaries + Total heat generated inside $[x,x+\Delta x]$
and writing the conservation equation
$\textit{Total Heat Inside} [x,x+\Delta x]= cpA \int _{ x}^{x+\Delta x}u(s,t) ds $
Takes derivative according to time, rewrites the equation
$ \frac{d}{dt} \int _{ x}^{x+\Delta x} c\rho A u(s,t) ds = c\rho A \int _{ x}^{x+\Delta x} u_t(s,t) ds $
$ = kA [ u_x(x+\Delta x,t) - u_x(x,t)] A \int _{x}^{x+\Delta x} f(s,t) ds $
At this point he wants to get rid of the integrals, so he uses the mean value theorem which is, for a $a < \xi < b$
$ \int _{ a}^{b} f(x) dx = f(\xi)(b-a) $
A $\xi$ must exist within the specified interval. He applies it to the equation, and gets
$ c\rho A u_t(\xi_1,t)\Delta x = kA[u_x(x+\Delta x, t) - u_x(x,t)] + Af(\xi_2,t)\Delta x $
$ x < \xi < x+\Delta x $
This also makes sense, there are multiple $\xi$'s for two different integrals. However below, he turns two $\xi$'s into one,
$ u_t(\xi,t) = \frac{k}{c\rho} \bigg[ \frac{u_x(x+\Delta x,t) - u_x(x,t)} {\Delta x} \bigg] + \frac{ 1}{c\rho}f(\xi,t) $
and while $ \Delta x \to 0 $
he gets
$ u_t(x,t) = \alpha^2u_{xx}(x,t) + F(x,t) $
In this last statement, $\xi$'s are replaced by $x$. So I have three questions: First how did the author combine the two $\xi$'s, second, while $\Delta \to 0$, how he turned them into $x$. I guess it is understandable if the $\Delta x$ becomes infinitesimally small, than whatever's inside can only be $x$? That still does not explain the two $\xi$'s though.
Third question: Net flux of heat across boundary uses $u_x(x+\Delta x,t) - u_x(x,t)$, that is derivative according to space, and takes difference between two sectional endpoints. Why $u_x$? Shouldn't we use simply $u$ here?