I am having trouble understanding the proof of Theorem 3 in "Inequalities" by Hardy, Littlewood and Pólya. This theorem states that the -th mean approaches the geometric mean as approaches zero.
I have seen the following post which makes things a little clearer (albeit using instead of :
Why is the 0th power mean defined to be the geometric mean?
However, I still cannot determine why:
(a): as tends to zero,
I have a pretty solid grasp of limits, as well as the log and exp functions, but I have never really been taught anything substantial on big/little-O notation, in particular as the variable approaches zero. Could somebody point me towards a suitable proof of (a) and (b) above please.