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[MUSIC] Let's consider this limit. 
The lim (2x-x), as X → ∞ = ∞. 

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That's a true statement, but there's two 
totally different ways to think about 

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this statement, and it really hinges on 
the distinction between potential and 

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actual ∞. 
To have a potentially infinite pile of 

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fish is to have an endless supply of 
fish, as many fish as you'd like to have, 

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and that sense really goes will with how 
we're thinking about infinity in these 

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limits. 
To say the limit of 2x minus x equals 

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infinity, as x approaches infinity, is to 
say that I can make 2x minus x as big as 

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I like, as long as x. 
Big enough. 

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Contrast that with actual infinity. 
To have an actually infinite pile of fish 

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would be to have, right now, a pile of 
fish that contains infinitely many fish 

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at this very moment. 
Is it possible to combine that way of 

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thinking with infinity, with these kind 
of limit statements? Let me share with 

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you a fable, to see one of the paradoxes 
that results. 

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Once upon a time, there was a house. 
I lived in that house, with my cat. 

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And we lived, near a lake, and this lake 
is full of fish, so every day we went 

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fishing. 
And each day I caught two fish. 

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The first day I caught fish labelled 1 
and 2. 

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Now, my cat prefers eating the lowest 
number of fish in our stockpile, so my 

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cat ate the fish numbered 1. 
The next day, I went fishing again, and I 

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caught fish labelled 3 and 4. 
And my pet, still preferring to eat the 

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lowest numbered fish. 
Eats fish number 2. 

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Another day another fishing expedition, I 
go fishing again, I get 2 more fish, 

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label 5 and 6, my cat preferring to eat 
the lowest number fish in our stock pile 

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eats fish number 3. 
I go fishing the next day and I get 2 

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more fish, fish label 7 and 8, my cat 
preferring to eat the lowest numbered 

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fish in our stock pile eats fish number 
4. 

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And so it goes forever. 
Each day our stockpile gets bigger. 

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There's more fish in my pile every single 
day. 

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And yet, at the end of time, do any fish 
remain. 

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On the Nth day my cat ate the Nth fish. 
So which fish survives my cat's appetite? 

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Human beings want to understand infinity, 
but reasoning about actual infinity is 

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liable to walk us straight into those 
kinds of paradoxes. 

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Instead, limits by focusing our attention 
on potential infinity provide a way to 

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reason about infinity that avoid those 
kinds of paradoxes. 

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[MUSIC] It provides a way for mere human 
beings to think about infinity in a 

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precise way. 
[MUSIC]