I am having trouble understanding the proof of Theorem 3 in "Inequalities" by Hardy, Littlewood and Pólya. This theorem states that the $r$-th mean approaches the geometric mean as $r$ approaches zero.

I have seen the following post which makes things a little clearer (albeit using $o(r)$ instead of $O({r}^{2})$:

Why is the 0th power mean defined to be the geometric mean?

However, I still cannot determine why:

(a): ${a}^{r}=1+r.log(a)+o(r)$ as $r$ tends to zero,

and

(b): $\underset{r\to 0}{lim}(1+rx+o(r){)}^{1/r}={e}^{x}$

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.