Limits.

I couldn't make it to the lecture for this...how do the underlined parts equate and is that supposed to be a zero? Could someone help please?

Thanks in advanced.

First underlined part: $f(x) = e^{\ln{f(x)}}$.

Second underlined part: Maclaurin series expansion of $e^{x}$. The $O(x^2)$ (it's not a 0, it's a capital O) part describes the error function I think - as $x\to 0$ i.e. gets infinitesimally small, all the terms with large powers of $x$ become very small so the significant terms are written out in the series expansion, and the rest of the terms are summarised with this $O(x^2)$.

First underlined part: $f(x) = e^{\ln{f(x)}}$.

Second underlined part: Maclaurin series expansion of $e^{x}$. The $O(x^2)$ (it's not a 0, it's a capital O) part describes the error function I think

as $x\to 0$ i.e. gets infinitesimally small, all the terms with large powers of $x$ become very small so the significant terms are written out in the series expansion, and the rest of the terms are summarised with this $O(x^2)$.

I'd be careful with terminology here: "the error function" has a specific meaning which is rather different - although context makes it fairly clear what you mean here, something like "Remainder term" would be more easily understandable here.

Trying to be accurate but still informal, one way to think of what O(x^2) means here is "some function of x that is no larger than a fixed multiple of x^2". The point being, "it doesn't really matter about the details, because any fixed multiple of x^2 is so small we can ignore it later".