There is certainly a typo and it should be $\Delta x\rightarrow 0$. The result being quoted is the integral mean value theorem. What's going on is as follows. In (6), they derive that the probability equals $f(x_0,y_0)\Delta x\Delta y$ for $x_0\in (x,x+\Delta)$ and $y_0\in (y,y+\Delta)$. This is precisely what the integral mean value says, that the integral of a continuous function is equal to the area on which you are integrating multiplied by the value of the function somewhere in that area (exactly where is somewhat immaterial as we'll see below). It's a sort of average value result.
The point is that in $(7)$, you are no longer looking just at $x_0,y_0$ but $f(x,y)$. But, if we interpret the mean value theorem as in $(6)$ then we realize that as $\Delta x$ and $\Delta y$ go to 0, then $x_0\rightarrow x$ and $y_0\rightarrow y$, so by continuity of $f$, we get $f(x_0,y_0)\rightarrow f(x,y)$. Again, here $x_0$ and $y_0$ depend both on $x,y$ and $\Delta x$, $\Delta y$, but we are guruanteed they lie somewhere in $(x,x+\Delta x)$ and $(y,y+\Delta y)$ respectively so as the intervals shrink to zero the $x_0$ and $y_0$ are squeezed arbitrarly close to $x,y$.