[MUSIC] Let's consider this limit. 
The lim (2x-x), as X → ∞ = ∞. 
That's a true statement, but there's two 
totally different ways to think about 
this statement, and it really hinges on 
the distinction between potential and 
actual ∞. 
To have a potentially infinite pile of 
fish is to have an endless supply of 
fish, as many fish as you'd like to have, 
and that sense really goes will with how 
we're thinking about infinity in these 
limits. 
To say the limit of 2x minus x equals 
infinity, as x approaches infinity, is to 
say that I can make 2x minus x as big as 
I like, as long as x. 
Big enough. 
Contrast that with actual infinity. 
To have an actually infinite pile of fish 
would be to have, right now, a pile of 
fish that contains infinitely many fish 
at this very moment. 
Is it possible to combine that way of 
thinking with infinity, with these kind 
of limit statements? Let me share with 
you a fable, to see one of the paradoxes 
that results. 
Once upon a time, there was a house. 
I lived in that house, with my cat. 
And we lived, near a lake, and this lake 
is full of fish, so every day we went 
fishing. 
And each day I caught two fish. 
The first day I caught fish labelled 1 
and 2. 
Now, my cat prefers eating the lowest 
number of fish in our stockpile, so my 
cat ate the fish numbered 1. 
The next day, I went fishing again, and I 
caught fish labelled 3 and 4. 
And my pet, still preferring to eat the 
lowest numbered fish. 
Eats fish number 2. 
Another day another fishing expedition, I 
go fishing again, I get 2 more fish, 
label 5 and 6, my cat preferring to eat 
the lowest number fish in our stock pile 
eats fish number 3. 
I go fishing the next day and I get 2 
more fish, fish label 7 and 8, my cat 
preferring to eat the lowest numbered 
fish in our stock pile eats fish number 
4. 
And so it goes forever. 
Each day our stockpile gets bigger. 
There's more fish in my pile every single 
day. 
And yet, at the end of time, do any fish 
remain. 
On the Nth day my cat ate the Nth fish. 
So which fish survives my cat's appetite? 
Human beings want to understand infinity, 
but reasoning about actual infinity is 
liable to walk us straight into those 
kinds of paradoxes. 
Instead, limits by focusing our attention 
on potential infinity provide a way to 
reason about infinity that avoid those 
kinds of paradoxes. 
[MUSIC] It provides a way for mere human 
beings to think about infinity in a 
precise way. 
[MUSIC]