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[MUSIC] So now we learn what the official 
definition of limit is. 

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To say that the limit of f(x) = L as x 
approaches a, means the following. 

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It means that for all epsilon bigger than 
zero, this backwards three is the real 

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number epsilon or the Greek letter, a 
variable. So for all epsilon greater than 

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zero, there's a delta greater than zero, 
this is the Greek letter delta. 

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So, for all epsilon greater than zero, 
there's a delta greater than zero. 

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So that if, this, if the absolute value 
of x - a is between zero and delta, then, 

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the absolute value of f(x) - L is less 
than epsilon. 

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When you say it like that, I think it's 
really hard to see how this has any 

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relationship to what a more intuitive 
description of this limit statement might 

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be. 
I mean, what's this trying to get at? 

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It's trying to say f(x) is as close as I 
want to L by making x sufficiently close 

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to a. 
So, how, how to reconcile those, those 

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two perspectives, right? How does this 
have anything to do with things being 

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close. 
The key, take a look at this absolute 

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value of the difference, 
right? 

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The absolute value of x - a is the 
distance between x and a. 

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So to say that the distance between x and 
a is between zero and delta is to say 

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that x is within delta of a, alright? The 
distance from x to a is less than delta. 

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And to say that the distance between x 
and a is bigger than zero is just to say 

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that the distance between x and, and a, 
you know, isn't zero, right, x isn't a. 

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So I can rewrite that, maybe in a little 
bit easier way. 

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[SOUND] So instead of saying that, it's 
the same thing to say, if x is not equal 

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to a, so the absolute value of x - a 
isn't zero, and x is within delta of a. 

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So the distance between x and a is delta. 
And I can do the same thing to this 

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absolute value of a difference, alright? 
The absolute value of f(x) - L, that's 

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the distance, between f(x) and L. 
And the sum of the distance between f(x) 

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and L is less than epsilon. 
Well, that just means that f(x) is within 

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epsilon of L. 
So, I'll rewrite that as that. [SOUND] 

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Here we go. 
[SOUND] Then f(x) is within epsilon of L. 

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So, I think when, when you write it like 
this, it makes a little bit more sense, 

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right? 
To say that the limit of f(x) = L as x 

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approaches a, 
means that for all numbers epsilon, 

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epsilon is measuring how close I want 
f(x) to L, 

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then, there's some corresponding number 
delta, which is how close x has to be to 

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a. 
So that whenever x is that close, delta, 

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within delta of a, 
then f(x) is really within epsilon of L. 

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And to say that the limit of f(x) = L 
means that no matter which epsilon I 

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choose, there's some corresponding delta, 
so that whenever x is within delta of a, 

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then f(x) is within epsilon of L. 
Now, how this actually gets played out 

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in, in more concrete situations can be, 
you know, kind of complicated, but this 

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is really the official definition of what 
it means to say that the limit of f(x) 

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equals L as x goes to a. [MUSIC] And 
we're going to be trying to unpack this 

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definition to see what it might mean in 
some specific cases.