PLEASE BEFORE I SAY ANYTHING NO LINKS AND I MEAN IT
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means to say that for any given , we can find
such that anytime
(i.e. the whenever
is "close enough" to 5), we can guarantee that
(i.e. the value of
is "close enough" to the limit value).
What we want to end up with is
Dividing both sides by 3 gives
which suggests is a sufficient threshold.
The proof itself is essentially the reverse of this analysis: Let be given. Then if
and so the limit is 7. QED