What does the limsup and liminf of functions tell us?

The limsup as x-> a of a function f from a metric space to R is

lim epsilon -> 0 [sup{f(x) : x in E intersect B(a,epsilon) \ {a} }]

Wikipedia has it written using latex https://en.wikipedia.org/wiki/Limit\_inferior\_and\_limit\_superior#Functions\_from\_topological\_spaces\_to\_complete\_lattices.

I don't really have a good intuition for limsup and liminf of a function like I do for sequences. It sounds like their difference is meaningful because Wikipedia says limsup - liminf at a point is defines the oscillation at that point.

Are they also useful on their own (just the limsup or just the liminf)? What sort of information can we get from them and what is a nontrivial example where lim =/= liminf=/= limsupof a function?

Also, why do we exclude the point {a} in the definition? Is this because if we include it then the limsup and liminf would just be equal to that point?