Remark
41
[20G](Solved on 2022-11-29) Using the properties of \(\inf ,\sup \), we obtain for example these characterizations
\[ \limsup _{xβ x_ 0} f(x) β€ l \iff β z{\gt}l, \text{ eventually for } xβ x_ 0, f(x){\lt} z ~ ~ ; \]
\[ \limsup _{xβ x_ 0} f(x) β₯ l \iff β z{\lt}l, \text{frequently for } xβ x_ 0, f(x){\gt} z ~ ~ ; \]
and so on. (In this simplified writing, we assume that \(xβ I\)).