I understand, or at least think I understand, the nature of a function that is "little o": If $f$ is a function between Banach spaces E and F, then it is "little-o" if
$|x|\rightarrow 0 \implies \frac{|f(x)|}{|x|} \rightarrow 0$
Thus the evaluation of $f$ at $x$ approaches $0$ faster than $x$ itself. I have read other posts on here, such as this one that give a different definition. Also, textbook authors don't seem to be in agreement either. For instance, in their advanced calculus text, Loomis and Sternberg declare a function to be "little o" if it satisfies essentially the definition I just gave but also add the condition that $f(0) = 0$. On the other hand, Marsden et. al. in "Manifolds, Tensor Analysis and Applications" define a "little o" function as any continuous function $f:E\rightarrow F$ such that $ \lim_{x\rightarrow 0}\frac{f(x^k)}{|x|^k} = 0 $
Is there any hope of reconciling these definitions? They seem to be saying approximately the same thing, but not quite.