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AP Calc: LIM‑1 (EU), LIM‑1.E (LO), LIM‑1.E.2 (EK)

We're now going
to think about one of my most favorite
theorems in mathematics, and that's the squeeze theorem. And one of the reasons that it's
one of my most favorite theorem in mathematics is that it
has the word "squeeze" in it, a word that you
don't see showing up in a lot of mathematics. But it is appropriately named. And this is oftentimes also
called the sandwich theorem, which is also an appropriate
name, as we'll see in a second. And since it can be called
the sandwich theorem, let's first just
think about an analogy to get the intuition behind
the squeeze or the sandwich theorem. Let's say that there
are three people. Let's say that there is
Imran, let's say there's Diya, and let's say there is Sal. And let's say that
Imran, on any given day, he always has the fewest
amount of calories. And Sal, on any
given day, always has the most number of calories. So in a given day,
we can always say Diya eats at least
as much as Imran. And then we can say Sal eats
at least as much-- that's just to repeat those
words-- as Diya. And so we could set up a
little inequality here. On a given day, we could
write that Imran's calories on a given day are going to be
less than or equal to Diya's calories on that same day,
which is going to be less than or equal to Sal's
calories on that same day. Now let's say that it's Tuesday. Let's say on
Tuesday you find out that Imran ate 1,500 calories. And on that same day, Sal
also ate 1,500 calories. So based on this,
how many calories must Diya have eaten that day? Well, she always eats at least
as many as Imran's, so she ate 1,500 calories or more. But she always has less than or
equal to the number of calories Sal eats. So it must be less
than or equal to 1,500. Well, there's only
one number that is greater than or equal
to 1,500 and less than or equal to 1,500, and
that is 1,500 calories. So Diya must have
eaten 1,500 calories. This is common sense. Diya must have had
1,500 calories. And the squeeze
theorem is essentially the mathematical version
of this for functions. And you could even view this is
Imran's calories as a function of the day, Sal's calories
as a function of the day, and Diya's calories as
a function of the day is always going to
be in between those. So now let's make this a
little bit more mathematical. So let me clear this out
so we can have some space to do some math in. So let's say that we
have the same analogy. So let's say that we
have three functions. Let's say f of x
over some interval is always less
than or equal to g of x over that same interval,
which is always less than or equal to h of x over
that same interval. So let me depict
this graphically. So that is my y-axis. This is my x-axis. And I'll just
depict some interval in the x-axis right over here. So let's say h of x looks
something like that. Let me make it more interesting. This is the x-axis. So let's say h of x looks
something like this. So that's my h of x. Let's say f of x looks
something like this. Maybe it does some interesting
things, and then it comes in, and then it goes up
like this, so f of x looks something like that. And then g of x,
for any x-value, g of x is always in
between these two. And I think you see where
the squeeze is happening and where the
sandwich is happening. If h of x and f of x were
bendy pieces of bread, g of x would be the meat of the bread. So it would look
something like this. Now, let's say that we know--
this is the analogous thing. On a particular day, Sal and
Imran ate the same amount. Let's say for a
particular x-value, the limit as f and h
approach that x-value, they approach is the same limit. So let's take this
x-value right over here. Let's say the x-value
is c right over there. And let's say that
the limit of f of x as x approaches c is
equal to L. And let's say that the limit as x approaches
c of h of x is also equal to L. So notice, as x
approaches c, h of x approaches L. As x approaches
c from either side, f of x approaches L. So these limits
have to be defined. Actually, the
functions don't have to be defined at x approaches c. Just over this
interval, they have to be defined as we approach it. But over this interval,
this has to be true. And if these limits right
over here are defined and because we know that g
of x is always sandwiched in between these two
functions, therefore, on that day or
for that x-value-- I should get out of that
food-eating analogy-- this tells us if all of this
is true over this interval, this tells us that the limit
as x approaches c of g of x must also be equal to L. And once again, this
is common sense. f of x is approaching
L, h of x is approaching L, g of x is sandwiched
in between it. So it also has to
be approaching L. And you might say, well,
this is common sense. Why is this useful? Well, as you'll see, this
is useful for finding the limits of some
wacky functions. If you can find a function
that's always greater than it and a function that's
always less than it, and you can find the limit
as they approach some c, and it's the same
limit, then you know that that wacky
function in between is going to approach
that same limit.

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