Theorem. Let f be a bounded function for x near a and ≠ a. Then
lim
x→ a
f(x) exists
if and only if
liminf
x→ a
f(x)
=
limsup
x→ a
f(x)
Proof. Easy - If limx→ af(x) = L, then liminfx→ af(x) = L, …. For the converse, use the first of the two
conditions in the characterizations of limsupx→ af(x) = L
and liminfx→ af(x) = L.
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version 4.05. On 18 Sep 2014, 21:28.