Where are the limits being taken in these thermodynamics equations?

[zenterix]

TL;DR Summary

In a passage in the book Heat and Thermodynamics by Zemansky, the notation omits the variable for which limits are being taken. I would like to understand the limits better.

Here is a passage from a book I am reading

My question is about the limits.

Are all the limits in the derivation above done for $P_{TP}\to 0$?

In particular, is it $\lim\limits_{P_{TP}\to 0} (Pv)$ that appears above?

The author omits this information in all but the first limit and it got me confused.

Here is a bit more context now to show why this has me confused.

Just before the equations above, the book writes of the fact that if we plot $Pv$ against $P$ for different gases at a specific temperature, we see that for all of the gases the limiting value of $Pv$ as $P\to 0$ is the same.

Here is an example at the boiling point of water

Here is my attempt at explaining away the confusion

The ideal-gas temperature definition involves a limit in which we compute the value of $P/P_{TP}$ as $P_{TP}$ is made to approach zero at constant volume.

The way I understand this, a constant volume pressure thermometer is used. We have some particular temperature that we would like to measure, for example that of steam.

Now, in order to make $P_{TP}$ smaller, in each successive measurement we have the same volume of gas in the thermometer but we remove some gas from the thermometer: this way, the triple point of water is reached at a lower pressure for the same constant volume.

As we make these successive measurements, I think that the pressure $P$ associated with the steam will also be lower and will approach zero just like $P_{TP}$ (even though the ratio of these two pressures will approach a non-zero value).

Thus, it seems that $\lim\limits_{P\to 0} (Pv)$ is the same as $\lim\limits_{P_{TP}\to 0} (Pv)$.

Is this what is happening?