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Limits: Examples Part 2 Video Lecture | Crash Course for MAT
FAQs on Limits: Examples Part 2 Video Lecture - Crash Course for MAT
| 1. What is a limit in mathematics? |  |
Ans. In mathematics, a limit is a fundamental concept that describes the behavior of a function as its input values approach a certain value. It determines the value that a function approaches as the input gets arbitrarily close to a particular point.
| 2. How is the limit of a function defined? | |
Ans. The limit of a function is defined as the value that the function approaches as the input approaches a particular value or as it tends to infinity or negative infinity. It is denoted using the symbol "lim" followed by the function and the variable approaching the limit.
| 3. Can limits be used to determine the continuity of a function? | |
Ans. Yes, limits play a crucial role in determining the continuity of a function. A function is said to be continuous at a point if the limit of the function exists at that point and is equal to the value of the function at that point. If the limit does not exist or is not equal to the value of the function, then the function is considered discontinuous at that point.
| 4. Are limits useful in solving real-life problems? | |
Ans. Yes, limits are widely used in solving real-life problems across various fields such as physics, engineering, economics, and computer science. They help in modeling and analyzing the behavior of systems, calculating rates of change, determining maximum and minimum values, and understanding the concept of infinity.
| 5. How can limits help in understanding the behavior of a function near a specific point? | |
Ans. Limits provide valuable insights into the behavior of a function near a specific point. By evaluating the limits, we can determine if the function approaches a finite value, diverges to infinity, oscillates, or exhibits other characteristic behaviors. This information helps in understanding the overall nature and properties of the function in its vicinity.
Text Transcript from Video
hello friends this video limits and
derivatives part four is brought to you
by exam feel calm no more fear from exam
before watching this video please make
sure that you have watched part 1 2 & 3
it's Tacoma example here we have this
function X purpose X we have to find
this limit at X almost equal to 3x
tending to click so this case let's
again we plot the graph that's a pulse
is 0 this is 1 this is 2 this is 3 4 so
X is going to 0 you get 0 x is going to
1 you get 1 plus 1/2 x go to 2 you get
this is f of 1 is equal to 1 plus 1 is
equal to 2 F of 2 is equal to 2 squared
plus 2 and this focus to 6 F of 3 is
equal to 3 squared plus 3 that is 9 plus
3 12 and F of 4 is equal to 4 squared
plus 4 that is straight you see these
values are increasing exponentially so
we can say that this is 0 and we'll
assume that this is 25 10 TT will assume
that this is 5 this is 10 15 this is 20
so we see F of 4 is 22 this point is 20
F of 3 is 12 almost new at this point
this 15 here snoring and F of 2 is 6
almost this point F of 1 is 1 almost on
this one and F of minus 1 will also be 4
minus 1 will be F of 0 is 0 and F of
minus 1 if you put F of minus 1 that
will become 1 square plus 1 that is zip
1 square minus 1 is 0 so how minus 1 is
also
zero this is minus 1 this Dell is also 0
F of minus 2 is equal to minus 2 square
that is 4 minus 2 that is 2 so this is
equal to 225 mo value F of minus 3 is
equal to minus 3 squares go to 9 minus 3
that is 6 so this guy is so we have
graphed something like this I will draw
the graph the graph will get will be
something like this something like this
this is the graph to get correct now we
have to find it X is equal to 3 so X is
equal to 3 czar this point that is
nothing but well this is our point now
this is a point in approach this point
on the left hand side so when I say
limit of X is equal to 3 minus of FX
that is nothing but f of X supposed to
0.99 correct because I am taking a value
that is near to 3 but less than 3 this
point somewhere here you're agreeing
here to 3 that is 2 point 9 9 9 so that
is nothing but 2 point 9 9 9 9 square
plus 2 point 9 9 and this value
calculate you get something like eleven
point nine nine nine something something
of this sort and that is almost equal to
similarly when you talk about limit of X
of 3 plus there is any number that is
near to 3 and it isn't three not three
so that is something of this song three
point zero zero zero one all right so
this number if you see this value is
nothing but three point zero zero one
square plus three point zero zero zero 1
and this value is of this form to help
find zeros
seven something of this form because
when you get the act actual value but
this will be of this form and this value
is almost equal to what a hair until um
we have first parted the graph nature
various values of X we got wise and then
we have found that limit of FX where X
limits three minus is almost well and
limits of FX X 3 plus is almost twelve
so now we observe that here also left is
equal to right limit this left limit is
equal to right limit so we can say that
limit X is tending to three of FX here
also it exists and that value is nothing
but and that is correct what we have
done via first part of the graph and he
found limit of x3 - and we have found
limit of X 3 plus and we observe that
both of others are always equal to 1
thus the concluded that since left limit
and right limit early are equal so the
limit at X equal to 3 for this function
FX exists and that value is 12 we take
our example we have to find the left
limit right limit and the limit at X is
going to 0 for this function for any
number that is less than 0 the function
value is 1 and for any number greater
than equal to 0 the function values - so
this 0 is included here so the function
value is any number less than equal to 1
is 1
this is for any number less than equal
to 1
please note 0 is included here and any
number greater than 0 that is this part
here is not I mean this 0 is not
including any number greater than 0 to
the 0 point 1 1 me number this values -
this is 1 this is 2 this is 3 all right
so if you see at X is equal to 0 the
value is 1 at X is equal to 0 point 0 0
1 the value is true now let's find a
left
so limit at X tends to zero - we are
approaching this function from the
negative direction this is isolate
direction so we see that this is nothing
but anything less than zero is 1 right
that is nothing but anything less than
zero we can say is -
0.9999 correct
this is value less than 0 and this value
is nothing but 1 similarly if you want
to find the value of this limit for X 0
+ that is the right limit that is
nothing but f of 1 point 0 0 0 1 correct
and F of 1 point 0 0 1 any number that
is greater than 0 is this that is it
will repeat but I have done limit 0 - is
any number less than 0 that is 0.9999 -
and that value is 1 because f of any
number that is less than 0 is 1 so f of
minus point nine nine nine nine will
also be 1 similarly limit of X 0 plus we
can take this value one point zero zero
zero one or point zero zero one sorry
because you need a mother is greater
than zero and almost equal to zero is
point zero zero zero one so f of point
zero zero one they lose equation because
this number is greater than zero and
then well is two so here we have seen
that in first case we got 1 and second
case in part two so here we say that
left limit is not equal to right limit
since left limit is not equal to right
limit in this case we say that limit X
tends to 0 for FX does not exist
please note here the limit at X is equal
to 0 for FX or something since they
cause the left limit is not equal to
right limit the left limit is equal to 1
and the right limit is 4 - this is the
one example where there is where FX
where X tends to 0
is not defined here we have defined left
limit right limit and limit at X is
equal to zero for this function and this
function the value of FX is equal to 1
or any number that is less than 0 and FX
is going to do for any number to the
zero so let's suppose this is 0 this is
1 this is 2 this 0 for X is equal to 0
and less than zero why is it going to 1
all this so this is the from here to
here and for any number that is greater
than 0 so it is not included the value
of y is 2 this is the typical graphical
representation of this equation FX is
equal to 1 for any number equal to 0 or
less than 0 so this at 0 point this
value is 1 and FX is you've got to do 2
for any number greater than 0 now let's
find the limit so in case of limit X is
0 minus left limit of FX that will be
any number it is less than 0 and almost
near to 0 that will be F minus 0.002
take an F of minus point 0 1 if you see
will be somewhere this point so for this
point the Y is 1 so this value is
nothing but the value of this function
at X equal to point zero minus point
zero zero zero one is bed to 1 similarly
let's find the right in the right hand
limit x is 0 plus of FX this anywhere
that is near to 0 and greater than 0
that is 0.0001 and take so f of any
number is greater than 0 this point is 2
you can see for this point if you see
the value is 1 and for any number which
is greater than 0 at this point the
value is 2
this is correct so here you see the left
limit is not equal to right limit why
because the left limit came out to be 1
and the right limit came out to be so
what is not equal to so left limit is
not equal to right level so we can say
that limit of this number at X is going
to 0 of this function is not defined
this the limit at X is equal to 0 for FX
is going to 5 in this case
thank you visit exam viacom to watch
free education videos try free on my
debts get the best quality study
materials study from the best tutors and
mentors and much more thanks again
derivatives part four is brought to you
by exam feel calm no more fear from exam
before watching this video please make
sure that you have watched part 1 2 & 3
it's Tacoma example here we have this
function X purpose X we have to find
this limit at X almost equal to 3x
tending to click so this case let's
again we plot the graph that's a pulse
is 0 this is 1 this is 2 this is 3 4 so
X is going to 0 you get 0 x is going to
1 you get 1 plus 1/2 x go to 2 you get
this is f of 1 is equal to 1 plus 1 is
equal to 2 F of 2 is equal to 2 squared
plus 2 and this focus to 6 F of 3 is
equal to 3 squared plus 3 that is 9 plus
3 12 and F of 4 is equal to 4 squared
plus 4 that is straight you see these
values are increasing exponentially so
we can say that this is 0 and we'll
assume that this is 25 10 TT will assume
that this is 5 this is 10 15 this is 20
so we see F of 4 is 22 this point is 20
F of 3 is 12 almost new at this point
this 15 here snoring and F of 2 is 6
almost this point F of 1 is 1 almost on
this one and F of minus 1 will also be 4
minus 1 will be F of 0 is 0 and F of
minus 1 if you put F of minus 1 that
will become 1 square plus 1 that is zip
1 square minus 1 is 0 so how minus 1 is
also
zero this is minus 1 this Dell is also 0
F of minus 2 is equal to minus 2 square
that is 4 minus 2 that is 2 so this is
equal to 225 mo value F of minus 3 is
equal to minus 3 squares go to 9 minus 3
that is 6 so this guy is so we have
graphed something like this I will draw
the graph the graph will get will be
something like this something like this
this is the graph to get correct now we
have to find it X is equal to 3 so X is
equal to 3 czar this point that is
nothing but well this is our point now
this is a point in approach this point
on the left hand side so when I say
limit of X is equal to 3 minus of FX
that is nothing but f of X supposed to
0.99 correct because I am taking a value
that is near to 3 but less than 3 this
point somewhere here you're agreeing
here to 3 that is 2 point 9 9 9 so that
is nothing but 2 point 9 9 9 9 square
plus 2 point 9 9 and this value
calculate you get something like eleven
point nine nine nine something something
of this sort and that is almost equal to
similarly when you talk about limit of X
of 3 plus there is any number that is
near to 3 and it isn't three not three
so that is something of this song three
point zero zero zero one all right so
this number if you see this value is
nothing but three point zero zero one
square plus three point zero zero zero 1
and this value is of this form to help
find zeros
seven something of this form because
when you get the act actual value but
this will be of this form and this value
is almost equal to what a hair until um
we have first parted the graph nature
various values of X we got wise and then
we have found that limit of FX where X
limits three minus is almost well and
limits of FX X 3 plus is almost twelve
so now we observe that here also left is
equal to right limit this left limit is
equal to right limit so we can say that
limit X is tending to three of FX here
also it exists and that value is nothing
but and that is correct what we have
done via first part of the graph and he
found limit of x3 - and we have found
limit of X 3 plus and we observe that
both of others are always equal to 1
thus the concluded that since left limit
and right limit early are equal so the
limit at X equal to 3 for this function
FX exists and that value is 12 we take
our example we have to find the left
limit right limit and the limit at X is
going to 0 for this function for any
number that is less than 0 the function
value is 1 and for any number greater
than equal to 0 the function values - so
this 0 is included here so the function
value is any number less than equal to 1
is 1
this is for any number less than equal
to 1
please note 0 is included here and any
number greater than 0 that is this part
here is not I mean this 0 is not
including any number greater than 0 to
the 0 point 1 1 me number this values -
this is 1 this is 2 this is 3 all right
so if you see at X is equal to 0 the
value is 1 at X is equal to 0 point 0 0
1 the value is true now let's find a
left
so limit at X tends to zero - we are
approaching this function from the
negative direction this is isolate
direction so we see that this is nothing
but anything less than zero is 1 right
that is nothing but anything less than
zero we can say is -
0.9999 correct
this is value less than 0 and this value
is nothing but 1 similarly if you want
to find the value of this limit for X 0
+ that is the right limit that is
nothing but f of 1 point 0 0 0 1 correct
and F of 1 point 0 0 1 any number that
is greater than 0 is this that is it
will repeat but I have done limit 0 - is
any number less than 0 that is 0.9999 -
and that value is 1 because f of any
number that is less than 0 is 1 so f of
minus point nine nine nine nine will
also be 1 similarly limit of X 0 plus we
can take this value one point zero zero
zero one or point zero zero one sorry
because you need a mother is greater
than zero and almost equal to zero is
point zero zero zero one so f of point
zero zero one they lose equation because
this number is greater than zero and
then well is two so here we have seen
that in first case we got 1 and second
case in part two so here we say that
left limit is not equal to right limit
since left limit is not equal to right
limit in this case we say that limit X
tends to 0 for FX does not exist
please note here the limit at X is equal
to 0 for FX or something since they
cause the left limit is not equal to
right limit the left limit is equal to 1
and the right limit is 4 - this is the
one example where there is where FX
where X tends to 0
is not defined here we have defined left
limit right limit and limit at X is
equal to zero for this function and this
function the value of FX is equal to 1
or any number that is less than 0 and FX
is going to do for any number to the
zero so let's suppose this is 0 this is
1 this is 2 this 0 for X is equal to 0
and less than zero why is it going to 1
all this so this is the from here to
here and for any number that is greater
than 0 so it is not included the value
of y is 2 this is the typical graphical
representation of this equation FX is
equal to 1 for any number equal to 0 or
less than 0 so this at 0 point this
value is 1 and FX is you've got to do 2
for any number greater than 0 now let's
find the limit so in case of limit X is
0 minus left limit of FX that will be
any number it is less than 0 and almost
near to 0 that will be F minus 0.002
take an F of minus point 0 1 if you see
will be somewhere this point so for this
point the Y is 1 so this value is
nothing but the value of this function
at X equal to point zero minus point
zero zero zero one is bed to 1 similarly
let's find the right in the right hand
limit x is 0 plus of FX this anywhere
that is near to 0 and greater than 0
that is 0.0001 and take so f of any
number is greater than 0 this point is 2
you can see for this point if you see
the value is 1 and for any number which
is greater than 0 at this point the
value is 2
this is correct so here you see the left
limit is not equal to right limit why
because the left limit came out to be 1
and the right limit came out to be so
what is not equal to so left limit is
not equal to right level so we can say
that limit of this number at X is going
to 0 of this function is not defined
this the limit at X is equal to 0 for FX
is going to 5 in this case
thank you visit exam viacom to watch
free education videos try free on my
debts get the best quality study
materials study from the best tutors and
mentors and much more thanks again
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