Welcome to a video on the Squeeze Theorem and Special Limits
Hopefully we can get through all of this in 10 minutes or less
the Squeeze Theorem states that if h of x is less than or equal to f of x
less than or equal to g of x for all x in an open interval containing c
except possibly at c itself
and if the limit as x approaches C of h of x
is equal to L which is equal to limit as x approaches c of g of x
so if both of these limits are equal to L
then the limit as x approaches c of f of x must also equal L
so graphically what's happening here is that
if we have h of x here and g of x here
where f of x is squeezed or sandwiched in between
these two functions
but in this case
if the limit of h and g as x approaches one is equal to two
we could also conclude that the limit as x approaches one of f of x is also equal to two
Let's consider the limit as x approaches zero of sine x divided by x
and see if we can figure out what this equal to using the Squeeze Theorem
so here's the setup, we're going to compare the area
of this large triangle to this sector
and to this smaller triangle where angle theta is on the unit circle
therefore this length is equal to one and this length will also equal one
this segment here would be the tangent segment
for this angle therefore this would be the height of this first
triangle member the area is equal to one-half
base times height well one-half times one
times tangent theta would of give us tangent theta divided by two
so that would be the area for this larger triangle here
now let's see if we can figure out the area for this sector here
and then lastly for this smaller triangle
here
the area of the sectors is one-half r theta
well since r is one we have one-half theta where theta divided by two
lastly on the small triangle here
this point here is on the unit circle so the y coordinate would be equal to sine theta
so the area would be one-half
times one time sine theta
or sine theta divided by two
geometrically we can see that the area of this sector is between the area of this larger triangle
and this smaller triangle
next what we're gonna do is multiply every term here by two divided by sine theta
notice I also changed tangent theta
into sine theta divided by cosine theta
now when we simplify
the result is one over cosine theta
greater than or equal to
theta divided by sine theta
which is greater than or equal to one
Let's go ahead and take this result onto the next screen
next we'll take the reciprocal of each of these terms so we have cosine theta over one
here we have sine theta
divided by theta and of course reciprocal of one is equal to one
now there's one catch here if we take the reciprocals of these and we have to reverse the inequality symbols
so now we would have less than or equal to in both cases
hopefully that makes sense because if you compare one-half and one-third for example we know one-half
is larger than one-third but if you take the reciprocals you have two and then three
to maintain a true statement you will have to reverse the inequality symbol there
now what we're going to do is take the limit of cosine theta and the limit of one
as theta approaches zero
well this limit would be equal to one
this isn't even affected by theta this is equal to one
so the limit, this is equal to one, limit of this is equal to one we can conclude using the Squeeze Theorem
that the limit
as theta approaches zero of sine theta
divided by theta must also equal one
okay there's two other special limits we want to discussed briefly we won't prove them
but we'll verify them
so here's the first one we just discussed next
the limit as x approaches zero of one minus cosine x divided by x is equal to zero
and then lastly the limit as x approaches zero of one plus x raised to the power of one divided by x is equal to
where there's number or e
Let's go ahead and take a look at the second special limit now
we're not gonna prove it but we'll go ahead and see if we can verify it
member there are three methods to determine a limit one of them is
numerically other one is graphically
Let's take at a look at the graph first one problem with using graphic technologies
it doesn't always graph
the functions accurately we know that this would be undefined at x equals zero
so there's actually a hole right here
however we can see that as we approach from the right and the left
we can see that function value is approaching zero
which is the value of this special limit
another way would be to look at the
look at the value of the function as x approaches zero here in the middle
So let's go ahead and take a moment and do that let's compare
the value of the function as x approaches zero
so we type in our function into Y1
let's go to our table set and change the independent variable to ask
go to our table and we just type in these values
so as you can see as we're approaching zero from the left
these are the function values let's go ahead and record those
and from the right we're going to have the same values just different signs
so here we have positive .05, positive .005
and you can see this is approaching zero so go ahead and accept this second special limit
last limit we want to look at is this one again if you take a look at the graph you can see
that this function would be undefined at x equal zero
so there would be a hole here
but we still can approach from the
left and the right so this limit is going to exist
you can see it's approaching the value that is less than three
member e is approximately 2.718
and let's go ahead and verify this with our table as well
so we'll go back and change our function
so Y1 we have our function let's go to our table
and you can see from the last values we have let's go ahead and put in the
positive values as well
that as x approaches zero it does look like it's approaching 2.718 or e
so we're go ahead and accept this special limit as well
again these are not proofs but this is a nice verification of the special limit
the next video we talk about how to find other limits using these special limits
thank you for watching