Class 11 Exam > Class 11 Videos > Example Sandwich Theorem - Limits & Derivatives
Example Sandwich Theorem - Limits & Derivatives Video Lecture - Class 11
FAQs on Example Sandwich Theorem - Limits & Derivatives Video Lecture - Class 11
| 1. What is the Sandwich Theorem in the context of limits and derivatives? |  |
Ans. The Sandwich Theorem, also known as the Squeeze Theorem, is a mathematical tool used in limits and derivatives. It states that if two functions, f(x) and g(x), both approach the same limit as x approaches a certain value, and a third function, h(x), is always between f(x) and g(x) for values of x near that point, then h(x) also approaches the same limit. In other words, if f(x) ≤ h(x) ≤ g(x) for all x near a certain value, and the limits of f(x) and g(x) as x approaches that value are both L, then the limit of h(x) as x approaches the same value is also L.
| 2. How is the Sandwich Theorem applied in real-life examples? | |
Ans. The Sandwich Theorem is often used to solve problems in physics, engineering, and other real-life scenarios. For example, it can be used to find the limit of a function that cannot be directly evaluated. By identifying two functions that "squeeze" the unknown function and have known limits, the Sandwich Theorem allows us to determine the limit of the unknown function. This is particularly useful when dealing with complicated functions or situations where direct computation is not feasible.
| 3. Can the Sandwich Theorem be used to evaluate derivatives? | |
Ans. No, the Sandwich Theorem is primarily used for evaluating limits, not derivatives. While the theorem helps determine the limit of a function, it does not provide any information about the function's derivative. Derivatives are calculated using different techniques, such as the power rule, product rule, or chain rule, depending on the specific function. The Sandwich Theorem is a tool specifically designed to analyze limits.
| 4. Are there any limitations to the Sandwich Theorem? | |
Ans. Yes, there are certain limitations to the Sandwich Theorem. One limitation is that it only applies when the functions f(x), g(x), and h(x) are defined and continuous near the point of interest. Additionally, the theorem requires that f(x) ≤ h(x) ≤ g(x) holds for all x near the point, which may not always be easy to prove. It is important to carefully assess the conditions and assumptions of the theorem before applying it to a specific problem.
| 5. Can the Sandwich Theorem be used to prove the existence of a limit? | |
Ans. Yes, the Sandwich Theorem can be used to prove the existence of a limit. If f(x) ≤ h(x) ≤ g(x) holds for all x near a certain point, and the limits of f(x) and g(x) as x approaches that point are both L, then the limit of h(x) as x approaches the same point also exists and is equal to L. This property of the Sandwich Theorem makes it a valuable tool in proving the existence of limits, especially when direct evaluation is challenging or not possible.
Text Transcript from Video
hello friends this video limits and
elevated part xi is walking by example a
calm no more fear from exam before
watching this video please make sure
that you have washed part 1 part then we
have to prove that kha'zix is always
less than sign x metrics and that for
all goes lesson for x is between 0 and
event so what you see in this this is a
circle this is a triangle is also a
triangle so if you see that the period
go a see if you take this mantle this
this and this fewer points every in this
triangle the area of this triangle will
always be less then the area of this
sector who seem to dot heavy this sector
which you get to see a deal correct and
this area with always be less then the
bigger area this one that is over VA you
can see that see if you see that this
value you see is this much and from this
this is the value c OC is this much
right so you can see on the all these 3
we have this this thing common this
thing this is OCA now on top of that if
you add this area are Keeler this is
again bigger or if an again on top of
that if you add this area this are going
to be net so we can say that area of
triangle or a/c is always bigger than in
your sector
Oh KC because the sector you got the
covariant and this is always greater
than we are 9/10 we all right because
OAB is a bigger triangle we can say
smaller triangle area is all this videos
is this sector because the sector
this area also and that is again always
bigger than the bigger triangle depth
against a more area private OAS is
nothing but 1 by 2 into oil into cdcd is
the height of this C and this is always
greater than now you have to find the
area of the sector so in your sector the
angle is X X degree so we'll convert
into Radian so X by 2 pi this is the
radiant in 2 pi you do okay into your
right PI R square this angle into PI R
square this angle is X pi 2 pi into our
service X is the Radian actually this X
is already not indeed so X by 2 pi 2 pi
is for the whole thing whew 16 there's a
formula of any sector which is a PI X
ready you formula is X by 2 pi into PI R
square by 2 way you into it and in your
this triangle wave is nothing burn
number 2 a into him all right now we see
that o is common in all so we can cancel
oasis is not 0 all right 1 by 2 is also
common in R will cancel this number T
here pi PI is for model cancer what we
are getting here is CD is always less
then takes into the wave this all this
Delta e now we have equations in the
term of cg.o in correct what we have to
find the equation in the form of course
X sine X X so if you can just find how
we can find this again write this C do a
Kobe in sine X cos x terms are probably
sort and we can very well do because we
know that in any triangle we know this
angle you can say example in this
triangle we can say that Kasich or tan X
is equal to CD by OD so we can do in
anytime
so this time will take
spangle or DC this triangle birthday in
this triangle if you want to find the
value of sine X sine X we know we know
the Spangenberg refresh our early very
go so sine X is P by H so P is CD and H
is forcing right so we can say that in
this triangle we write in triangle o CD
in triangle this will be you can say
that sine X is nothing but e SP is here
CD because this is the angle X this is
90 degree this is CD p a CD CD by H H is
all right now but we don't have o
see here we have CD yes we do have CD
here we have a adb you can see that this
OC is our oh is also our correct so we
can write Rho C is equal to or we know
that here was equal to one way is equal
to one because if both these are radius
so we can write this CD by Posey has
nothing but CD by coil so with that we
know that we have C do everything in
this solution so the first equation we
get here is right here is sine X is
going to CD by that is the first
equation right the second equation we
can get is in triangle over AP in this
triangle OAB if we want to find the
value of 10x but because we have a be
here a big load is coming tell me so in
triangle II you see 10 X to be AV by way
correctly and we have a B here we have
where so we will say that an X is equal
to AV y so we have equation now what we
can do is you can just substitute the
basket
so if you see CD is nothing but cosine X
so as you know CD all right
Oh a sign X X way I can leave it now
because I feel that way will get
canceled a B a B I can write as nothing
but mu V tan X or Y - Oh a 10x correct
re CE o is common in here so will cancer
way but so what we get here is sine X is
less than X and is less than tan X sorry
way part is cancer not that panics this
part is not destiny way bodies can see
this tan X tan X I get by - nothing but
sine X PI cos explain so let me write
this as this is nothing next X is less
then smell things man correct now we
divide the whole thing by sides because
we have sine X hairs in X - 9 X 8 and
the middle term should be X by sign it's
correct will divide the whole thing by
sine X so this becomes by sine X this
becomes X by sine X and this becomes 1
by just this cancel so what this little
castle becomes 1 so what we get is 1 is
less than X by sleeves and is less than
1 line that is what we have got L now
what we'll do we'll take a reciprocal we
take a reciprocal so if you take a
reciprocal 1 by cos x by 1 1 1 by 1 by
cos x this becomes cos x and this comes
this side so this is equal to this X by
sine X becomes sine X by X in one by
ones that is what we do this is what
what we have that we have your sandwich
theorem here so we have seen that in
this case our area triangle for you see
was less than
area of sector OAC and the
again lesser than area of triangle baby
we have found the value of all those
forms of the avium we found a and n by T
was commonly pants in that we have got
this equation in C D OE v term but we
wanted to find this equation sine X and
casas form so we have got this two
efficiency sine X is equal to C divided
into an excellent below it and we
replace this value of CD and AV here in
this equation so regard the sine X is
less than X is less than panics you have
written sign tan X as sine X by Cossacks
and we have divided the whole thing by
sine X we have got this equation and we
just we have taken the reciprocal to get
the answer that this kha'zix is always
there in sine X by X is always less than
using Santis theorem we have rules
here we have to prove that sine X where
X limit X tends to 0 is equal to 1 so
what we would look here here will say
will again use the formula Dishman
derived in the given example because if
you say here sine X by X X equal to 0 if
you put this becomes 0 y 0 that is 0 by
0 form so that is begin with something
for this you can try to write this in
the different form so we have this
derivation divide it cos x is always
less than smile expect X all this wisdom
one correct how do you say equal to also
because they can be a squared X equal to
0 and things will be PI by 2 this is the
you have to find this value for X values
so if you can prove that for X is going
to 0 this value example K for X going to
0 for X is equal to 0 and this goal is
also K so in that his visit this that is
also K can be still because we are told
that this cost X is always less than
equal to small X is longer this is equal
to 1 so you can prove that for X equal
to 0 this value is K this value is K is
same values then you can do that sine X
by X is all this
sweetie so for X equal to 0 cos of X is
nothing but cos of 0
all right and four X is equal to zero
one does all this one if you want right
is the limit form you can say that limit
or excuse me go because X is equal to
one limit or one is equal to one so you
see what the values are seen thus we can
move by sandwiched theory now I can now
assume by a sandwich theorem that sine X
Y X limit X tends to 0 is not including
right because we were told that sine X
is always less than equal to 1 and cos x
let me have found the balance of cos x
of X tends to 0 and 1 at X tends to 0
actually what the weather came out to be
1 and since the values are same they can
and Tom from sandwich theorem that sine
X by X disability 1/4 X 2 is 2 so the
example they have to prove that 1 minus
cos x by X limit extension so is going
to zip it first let's just put the value
0 what we get 1 minus cos 0 is 0 cos 0
is 1 as it is not in fine so 0 by 0 5
we can do something for this so what we
can do to rewrite this 1 minus cos x we
know is nothing but 2 sine squared X
button
you know this 1 minus cos it's nothing
but 2 sine square X bedrooms so we can
write this as 2 sine square expecting by
X
limit X tends to 0 correct or we can
write this as a from where sine of X by
2 into swing of this value then this
becomes PI expect this to and a crystal
right the whole thing limit X tends to
or I can write this as limit of X tends
to 0 smile is going to into limit of X
tends to 0 sine of X by by again write
this well because we know that the limit
of X tends to a or which equals P Q is
nothing but limit of X tends to a we
should be into limit of X tends to e
this window right same thing and use
limit of X tends to 0 or this values PQ
is equal to limit of X tends to 0 P
limit of X tends to 0 Q now this value
then put the value of 0 here so this
becomes sine of 0 by 2 into this value
we know that limit of sine X by 2 pi X
by 2 X 10 to 0 is equal to 1 we have saw
this in the last example that sine x by
x limit X tends to 0 is X tends to 0
again say X been friends you don't find
the same thing because X tends to 0 by
the same thing this becomes 1 this is
going to size 0 is 0 0 into 1 so this
value is very simple what we have done
we have just read it at 1 minus cos x f2
signs for X by 2 and then we have
rewritten this equation with this form
and we have used the formula limit of
sine X by X where X tends to 0 is 1 and
they've got the answer in physics thank
you
visit exam viacom to watch free
education videos try free online tests
get the best quality study materials
study from the best tutors and mentors
and much more thanks again
elevated part xi is walking by example a
calm no more fear from exam before
watching this video please make sure
that you have washed part 1 part then we
have to prove that kha'zix is always
less than sign x metrics and that for
all goes lesson for x is between 0 and
event so what you see in this this is a
circle this is a triangle is also a
triangle so if you see that the period
go a see if you take this mantle this
this and this fewer points every in this
triangle the area of this triangle will
always be less then the area of this
sector who seem to dot heavy this sector
which you get to see a deal correct and
this area with always be less then the
bigger area this one that is over VA you
can see that see if you see that this
value you see is this much and from this
this is the value c OC is this much
right so you can see on the all these 3
we have this this thing common this
thing this is OCA now on top of that if
you add this area are Keeler this is
again bigger or if an again on top of
that if you add this area this are going
to be net so we can say that area of
triangle or a/c is always bigger than in
your sector
Oh KC because the sector you got the
covariant and this is always greater
than we are 9/10 we all right because
OAB is a bigger triangle we can say
smaller triangle area is all this videos
is this sector because the sector
this area also and that is again always
bigger than the bigger triangle depth
against a more area private OAS is
nothing but 1 by 2 into oil into cdcd is
the height of this C and this is always
greater than now you have to find the
area of the sector so in your sector the
angle is X X degree so we'll convert
into Radian so X by 2 pi this is the
radiant in 2 pi you do okay into your
right PI R square this angle into PI R
square this angle is X pi 2 pi into our
service X is the Radian actually this X
is already not indeed so X by 2 pi 2 pi
is for the whole thing whew 16 there's a
formula of any sector which is a PI X
ready you formula is X by 2 pi into PI R
square by 2 way you into it and in your
this triangle wave is nothing burn
number 2 a into him all right now we see
that o is common in all so we can cancel
oasis is not 0 all right 1 by 2 is also
common in R will cancel this number T
here pi PI is for model cancer what we
are getting here is CD is always less
then takes into the wave this all this
Delta e now we have equations in the
term of cg.o in correct what we have to
find the equation in the form of course
X sine X X so if you can just find how
we can find this again write this C do a
Kobe in sine X cos x terms are probably
sort and we can very well do because we
know that in any triangle we know this
angle you can say example in this
triangle we can say that Kasich or tan X
is equal to CD by OD so we can do in
anytime
so this time will take
spangle or DC this triangle birthday in
this triangle if you want to find the
value of sine X sine X we know we know
the Spangenberg refresh our early very
go so sine X is P by H so P is CD and H
is forcing right so we can say that in
this triangle we write in triangle o CD
in triangle this will be you can say
that sine X is nothing but e SP is here
CD because this is the angle X this is
90 degree this is CD p a CD CD by H H is
all right now but we don't have o
see here we have CD yes we do have CD
here we have a adb you can see that this
OC is our oh is also our correct so we
can write Rho C is equal to or we know
that here was equal to one way is equal
to one because if both these are radius
so we can write this CD by Posey has
nothing but CD by coil so with that we
know that we have C do everything in
this solution so the first equation we
get here is right here is sine X is
going to CD by that is the first
equation right the second equation we
can get is in triangle over AP in this
triangle OAB if we want to find the
value of 10x but because we have a be
here a big load is coming tell me so in
triangle II you see 10 X to be AV by way
correctly and we have a B here we have
where so we will say that an X is equal
to AV y so we have equation now what we
can do is you can just substitute the
basket
so if you see CD is nothing but cosine X
so as you know CD all right
Oh a sign X X way I can leave it now
because I feel that way will get
canceled a B a B I can write as nothing
but mu V tan X or Y - Oh a 10x correct
re CE o is common in here so will cancer
way but so what we get here is sine X is
less than X and is less than tan X sorry
way part is cancer not that panics this
part is not destiny way bodies can see
this tan X tan X I get by - nothing but
sine X PI cos explain so let me write
this as this is nothing next X is less
then smell things man correct now we
divide the whole thing by sides because
we have sine X hairs in X - 9 X 8 and
the middle term should be X by sign it's
correct will divide the whole thing by
sine X so this becomes by sine X this
becomes X by sine X and this becomes 1
by just this cancel so what this little
castle becomes 1 so what we get is 1 is
less than X by sleeves and is less than
1 line that is what we have got L now
what we'll do we'll take a reciprocal we
take a reciprocal so if you take a
reciprocal 1 by cos x by 1 1 1 by 1 by
cos x this becomes cos x and this comes
this side so this is equal to this X by
sine X becomes sine X by X in one by
ones that is what we do this is what
what we have that we have your sandwich
theorem here so we have seen that in
this case our area triangle for you see
was less than
area of sector OAC and the
again lesser than area of triangle baby
we have found the value of all those
forms of the avium we found a and n by T
was commonly pants in that we have got
this equation in C D OE v term but we
wanted to find this equation sine X and
casas form so we have got this two
efficiency sine X is equal to C divided
into an excellent below it and we
replace this value of CD and AV here in
this equation so regard the sine X is
less than X is less than panics you have
written sign tan X as sine X by Cossacks
and we have divided the whole thing by
sine X we have got this equation and we
just we have taken the reciprocal to get
the answer that this kha'zix is always
there in sine X by X is always less than
using Santis theorem we have rules
here we have to prove that sine X where
X limit X tends to 0 is equal to 1 so
what we would look here here will say
will again use the formula Dishman
derived in the given example because if
you say here sine X by X X equal to 0 if
you put this becomes 0 y 0 that is 0 by
0 form so that is begin with something
for this you can try to write this in
the different form so we have this
derivation divide it cos x is always
less than smile expect X all this wisdom
one correct how do you say equal to also
because they can be a squared X equal to
0 and things will be PI by 2 this is the
you have to find this value for X values
so if you can prove that for X is going
to 0 this value example K for X going to
0 for X is equal to 0 and this goal is
also K so in that his visit this that is
also K can be still because we are told
that this cost X is always less than
equal to small X is longer this is equal
to 1 so you can prove that for X equal
to 0 this value is K this value is K is
same values then you can do that sine X
by X is all this
sweetie so for X equal to 0 cos of X is
nothing but cos of 0
all right and four X is equal to zero
one does all this one if you want right
is the limit form you can say that limit
or excuse me go because X is equal to
one limit or one is equal to one so you
see what the values are seen thus we can
move by sandwiched theory now I can now
assume by a sandwich theorem that sine X
Y X limit X tends to 0 is not including
right because we were told that sine X
is always less than equal to 1 and cos x
let me have found the balance of cos x
of X tends to 0 and 1 at X tends to 0
actually what the weather came out to be
1 and since the values are same they can
and Tom from sandwich theorem that sine
X by X disability 1/4 X 2 is 2 so the
example they have to prove that 1 minus
cos x by X limit extension so is going
to zip it first let's just put the value
0 what we get 1 minus cos 0 is 0 cos 0
is 1 as it is not in fine so 0 by 0 5
we can do something for this so what we
can do to rewrite this 1 minus cos x we
know is nothing but 2 sine squared X
button
you know this 1 minus cos it's nothing
but 2 sine square X bedrooms so we can
write this as 2 sine square expecting by
X
limit X tends to 0 correct or we can
write this as a from where sine of X by
2 into swing of this value then this
becomes PI expect this to and a crystal
right the whole thing limit X tends to
or I can write this as limit of X tends
to 0 smile is going to into limit of X
tends to 0 sine of X by by again write
this well because we know that the limit
of X tends to a or which equals P Q is
nothing but limit of X tends to a we
should be into limit of X tends to e
this window right same thing and use
limit of X tends to 0 or this values PQ
is equal to limit of X tends to 0 P
limit of X tends to 0 Q now this value
then put the value of 0 here so this
becomes sine of 0 by 2 into this value
we know that limit of sine X by 2 pi X
by 2 X 10 to 0 is equal to 1 we have saw
this in the last example that sine x by
x limit X tends to 0 is X tends to 0
again say X been friends you don't find
the same thing because X tends to 0 by
the same thing this becomes 1 this is
going to size 0 is 0 0 into 1 so this
value is very simple what we have done
we have just read it at 1 minus cos x f2
signs for X by 2 and then we have
rewritten this equation with this form
and we have used the formula limit of
sine X by X where X tends to 0 is 1 and
they've got the answer in physics thank
you
visit exam viacom to watch free
education videos try free online tests
get the best quality study materials
study from the best tutors and mentors
and much more thanks again
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Jan 10, 2025
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Example Sandwich Theorem - Limits & Derivatives for Class 112025 is part of Class 11 preparation. The notes and questions for Example Sandwich Theorem - Limits & Derivatives
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