Let's think geometrically.
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We're often exploring the limit of a
sequence by looking at some numeric evidence.
You know, we might evaluate a few hundred terms of
the sequence and notice, we're getting really close to something numerically.
And we might guess that that's the limit.
But we can also think about limits in a, in a more geometric way.
So, here I've got a number line, and I've
plotted the terms in my sequence on that number line.
And, it looks like the limit is L.
But what does that mean?
Well, if I zoom in on the number line a bit, it looks
like all of the terms of my sequence are within a hundredth of L.
As long as I'm after the 53rd term in my sequence.
And, if I
zoom in again. All the terms of my sequence are within a
thousandth of L, as long as I'm after the 181st term in my sequence.
And as close as I want to get to L, I can
do that, as long as I go far enough out in the sequence.
So a limit's really a promise.
And it's a promise that, if you want the terms of your sequence to be close to L,
well you can do that.
You can get the terms to be as close as you want to L, as long as you
throw away enough of the initial terms and just
restrict your attention to the tail of the sequence.
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