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Let's think geometrically.

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[MUSIC]

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We're often exploring the limit of a

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sequence by looking at some numeric
evidence.

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You know, we might evaluate a few hundred
terms of

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the sequence and notice, we're getting
really close to something numerically.

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And we might guess that that's the limit.

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But we can also think about limits in a,
in a more geometric way.

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So, here I've got a number line, and I've

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plotted the terms in my sequence on that
number line.

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And, it looks like the limit is L.

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But what does that mean?

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Well, if I zoom in on the number line a
bit, it looks

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like all of the terms of my sequence are
within a hundredth of L.

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As long as I'm after the 53rd term in my
sequence.

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And, if I

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zoom in again.
All the terms of my sequence are within a

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thousandth of L, as long as I'm after the
181st term in my sequence.

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And as close as I want to get to L, I can

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do that, as long as I go far enough out in
the sequence.

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So a limit's really a promise.

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And it's a promise that, if you want the
terms of your sequence to be close to L,

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well you can do that.

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You can get the terms to be as close as
you want to L, as long as you

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throw away enough of the initial terms and
just

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restrict your attention to the tail of the
sequence.

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[SOUND]