Yes that Hardy. The G H Hardy book is almost unobtainable at a sensible price, depending on the edition it goes fro between US$100 and US$500. But I do have a copy as I am trying to collect Hardy's works.

If you knew that

you know the limit form is true, and vice versa, which is why they are equivalent statements. You could arrive at

by a derivation by other means than the limit, so could deduce the limit from this. It only looks peculiar because you have not seen examples of the argument going both ways.

For instance as we know

we can deduce that

(we actualy need a bit more knowlege about the properties of

and monotonic increasing functions, but lets just suppose we have that knowlege for now), and so:

RonL