Main content

## Trigonometric limits & squeeze theorem

# Squeeze theorem example

## Video transcript

The graphs of f of x, g of x,
and h of x are shown below. Select and drag cards to
create a compound inequality that orders the
values of f of x, g of x, and h of x for x-values
near 2 but not at 2 itself. So for any of the x-values that
are depicted right over here, say, x is equal to 3, we see
that h of 3 is the largest, f of 3 is the smallest,
and g of 3 is in between. And that's true for
any of the x-values that they've depicted here. If we look at when x is equal
to 1, h of 1 is the largest, f of 1 is the smallest,
g of 1 is in between. So for all of the x-values
that they're depicting, f of x is less than or equal
to g of x, which is less than or equal to h of x. And the only place where they
equal based on this graph comes into play, it looks like
as we approach x equals 2, it looks like all the
functions are approaching 1. So that's where the equal
might come into play. But let's keep seeing what
they want us to do after that. So then it says it follows that. And instead of writing f
of x, g of x, and h of x, they've written the actual
definitions of them. So let's just remind
ourselves f of x is 2x times the square root of
x minus 1 minus 1. That's this blue
one right over here. So instead writing
f of x, we can write 2 times the square
root of x minus 1. Minus 1 is less than
or equal to g of x. G of x was this rational
expression right over here. So let's go back down here. We get this rational expression. And then that's going to
be less than or equal to h of x, which was, I believe,
e to the x minus 2. Is that right? Yep, e to the x minus 2. So all we've really
done is replaced f, g, and h with
their definitions. And then this means that
the limit-- so they're looking at the limit
as x approaches 2 of these different
expressions. So the limit as x approaches
2 of this expression is going to be less than
or equal to the limit as x approaches 2 of
this expression, which is this right over here,
which is going to be less than or equal to the
limit as x approaches 2 of this expression, which
is that right over there. And then they say
finally the value of the limit as x approaches 2
of this thing right over here is-- well, this is where
the squeeze theorem comes into play. We just have to remind
ourselves-- well, let's just think about it. Can we figure out the
limit as x approaches 2 of this right over here? Well, the limit as x approaches
2-- let's see, 2 minus 1. So we're taking the
principal root of 2 minus 1, which is the
principal root of 1. So you have 2 times 1 minus 1. So this is 1. This right over here
is e to the 2 minus 2. That's either the 0,
or that's 1 as well. So the limit of all
of this is going to be greater than
or equal to 1, and it's going to be
less than or equal to 1. Or it's right in
between 1 and 1. And the only way that it's
going to be between 1 and 1 is if it is equal to 1. This is the squeeze theorem
at play right over here. g of x, over the domain
that we've been looking at, or over the x-values
that we care about-- g of x was less than or equal
to h of x, which was-- or f of x was less than or equal
to g of x, which was less than or equal to h of x. And then we took the limit for
all of them as x approached 2. For the lower function, for
f of x, it approached 1. And we see it in the
graph right over here. In the lower function,
f of x approaches 1, h of x approaches 1,
and therefore, g of x must also approach 1. And we actually see that in
this graph right over here. But anyway, we could
check our answer just to feel good about ourselves. We got it right.