The book I am currently reading defined the big oh operator as the following: A function $ g (x) $ said to be $ O (h (x)) $ as $ x \to l $ if $\lim \sup_{x \to l} |g (x)/h (x)| < \infty $. What I don't get about this definition is the supremum.
First of all, using the definiton of limit supremum, can the above statement be interpreted as $\lim_{x \to l} \sup_{ |l - k| < x} |g (k)/h (k)| < \infty$ , meaning that we are taking the limit of the supremums of the $ g / h $ division on decreasing intervals around $ l $?
Secondly, why do we need the supremum? Why a straightforward limit like $\lim_{x \to l} |g (x)/h (x)| < \infty $ is not enough?
Thanks in advance