Commerce Exam > Commerce Videos > Mathematics (Maths) Class 11 > Limits: Examples Part 1
Limits: Examples Part 1 Video Lecture | Mathematics (Maths) Class 11 - Commerce
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FAQs on Limits: Examples Part 1 Video Lecture - Mathematics (Maths) Class 11 - Commerce
| 1. What is a limit in mathematics? |  |
Ans. In mathematics, a limit is a fundamental concept that describes the behavior of a function or a sequence as its input values approach a certain point. It represents the value that a function or sequence approaches as the input or index gets arbitrarily close to a specified value.
| 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 certain value. Mathematically, it can be represented as follows: lim(x->a) f(x) = L, where "lim" represents the limit, x is the input, a is the specified point, f(x) is the function, and L is the limit value.
| 3. What are the different types of limits? | |
Ans. There are several types of limits that can occur in mathematics. Some common types include the limit at a point, one-sided limits, and infinite limits. The limit at a point represents the behavior of a function as it approaches a specific value. One-sided limits describe the behavior of a function as it approaches from either the left or the right side of a point. Infinite limits occur when the function approaches positive or negative infinity as the input approaches a certain value.
| 4. How can limits be evaluated? | |
Ans. Limits can be evaluated using various techniques, depending on the function and the specific situation. Some common methods include direct substitution, factoring, rationalizing, and the use of special limits or rules such as the limit laws or L'Hôpital's rule. These techniques help simplify the function or remove any indeterminate forms, allowing for the determination of the limit value.
| 5. What is the significance of limits in calculus? | |
Ans. Limits play a crucial role in calculus as they provide the foundation for concepts such as continuity, derivatives, and integrals. The concept of a limit allows for the precise definition of these fundamental concepts, enabling mathematicians and scientists to analyze and solve various problems related to change, rates of change, and accumulation. Without limits, calculus would not exist in its current form and many of its applications in physics, engineering, and other fields would be impossible.
Text Transcript from Video
hello friends this video limits and
derivatives part 3 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 and
Martin now will take the example of
limit so these function is FX is going
to do and we have to find at 4 left hand
limit right hand limit any limit it
stalls Ram FX is going to always to you
do whatever you want but this gun is
always to this value will be something
like this this is the rather this is 1/2
stool this is 3 this is 4 this is 5 so
we have to find the left-hand limit at 4
right hand limit at 4 and the limit at
for example now if you see I mean to say
limit X 4 minus of X this is nothing but
left-hand limit so if you take any value
which is little less than 4 little less
than 4 pazam poof f of I'll take three
point nine nine nine nine
so f of three point nine nine but the
value do you put any other x you get to
only so f of three point nine days
similarly if you want to find the right
limit of this you will say limit X 4
plus of X that is equal to let's suppose
will take any whether that is greater
than phone and nearer to 4 there is 4
point 0 1 and this value is also because
you say any value of you see a given U
of X that is 2 this is the function you
take four point zero one you get two you
take three point nine nine nine you get
T's so here if you see the left is equal
to right correct the left limit is equal
to the right limit so we can say the
limit at X is equal to four exist since
the left is equal to right so we can say
that limit at X is equal to
they exist and the value of limit at X
is going to the forest by this since
this will be able to left in limit or
right limit in this case left right do
the same so this is this is what we have
done we have found the left limit and we
have found the right limit and we saw
that left and right there same that
means we can conclude that limit for
this function at X equal to 4 exists and
then we can say yeah this is the back we
take one more example if X is equal to X
square and then X is going to 4 we have
to find the left limit and right limit
let us draw this it's short once again
FX equal to X square we have to find the
limit left limit right limit and the
limit where X is almost equal to 4 so
let us draw the graph first let's say
this is 1 this is 2 this is 3 this is 4
5 6 so let's suppose the graph is
something like this X go to 0 so we have
equation y is equal to X square all
right X is going to 0 Y is equal to 0 X
is going to 1 Y is equal to 1 X equal to
2 y is equal to 2 squared - 4 X is equal
to 3 y is equal to 3 squares go to 9 so
let's draw this X going to 1 0 Y is
going to 0 X is equal to 1 Y is equal to
1 X is equal to 2 y is equal to 4 1 2
correct x equal to 3 y is equal to 9 so
my hair so we'll get a graph like this
actually so the graph will be something
like this it's it is the graph correct
so let's suppose this is 4 this is 4 X
good 2 3 y is equal to 9 that so this is
9 and X equal to 4 y is equal to 16
let's Appaloosa 16 this is the graph we
have similar go - house it will be
almost same it will be the mirror image
actually
mr. laugh now we have to find the left
limit and the right limit so let me put
a dot here at this point is equal to
four is this sorry this is extend is
wrap will extend this draw this is
sixteen actually this is three this is
nine okay now this is the wrap now this
is 16 X is going to 4 y is equal to 16
now the question is we have to find the
left limit left limit is you say limit X
is 4 - there is nothing where the value
of function that of FX is nothing but
value of function at 4 - it is three
point nine nine nine all right so what
is the value of function at 3.99 is
equal to three point nine nine nine
square all right say three point nine
nine square what you get is something
all fifteen point nine nine nine nine
this is where you get
similarly for right limit limit of X for
class of FX is nothing but f of four
point zero zero zero one correct that is
nothing but four point zero zero one
square because Y is equal to x squared
and that is nothing worth sixteen point
zero zero one six that is the value we
get almost that very we get so if you
say this fifteen point nine nine nine is
almost equal to 16 and 16 point zero
zero one is always almost equal to
sixteen so we can say that in this case
left limit is equal to right limit so
here you say left is equal to so we can
say that the limit also exists when I
say limit X tends to four we are not
used plus minus here of FX then that
value is so in this point at this point
four I am talking about left limit and
ever watching this
from this side I get value 16 when I
approve this point from this side right
side I get a low 16 so I say that limit
X equal to 4 or FX is nothing but 16 but
I have done here I have found the left
limit and the right limit and I observe
that left limit is equal to right limit
let's take one more example here we have
to find the limit for FX equal to X plus
1 where X is near to X limits to X is
almost equal to 2 so let's plot this
graph this is 1 this is 2 this is 3 this
is 4 and this is 0 so X go to 0 Y is
equal to 0 plus 1 1 X equal to 1 Y is
equal to 1 plus 1/2 it is going to do Y
is equal to 2 plus 1 3 X 3 y is what is
available so we get graph like this so
the graph to get something like this of
a straight line it should be a straight
line actually this is the graph again
now at X is going to to this point we
have to find the left limit and the
right limit so in case of left limit we
have to approve this point from this
side left side so when I say limit X
tends to 2 minus there is left limit of
f of X that is nothing but f of 1.999
let's of this value so what you get is
this is nothing but 1 point 9 9 9 9 plus
1 and this is nothing but 2 point 9 9 9
and that is almost equal to 3 so you see
that left limit of this function is 3
similarly now let me approach this
function from right side so when I say
limit of X of 2 class of FX that is
nothing but value of this function at 2
point 0 0 0 1 anywhere is near to 2 it
is greater than 2 so this is nothing but
2
zero zero zero zero one last one and
there is nothing but three point zero
zero zero one and this value is almost
equal to three so here you see the left
limit and the right limit both are equal
you see here left is equal to right and
that is equal to three so we can say
that in this case limit at X is equal to
4 FX exists limit for this exists and
that value is nothing but that is answer
very simple we have on the Left limit in
the right limit and we saw that with
left limit invite Lynda equal so we
throw on that left limit and right limit
are equal so limit for this function at
X equal to exists and that's value is 3
thank you visit exam viacom to watch
free education videos try free online
test
get the best quality study materials
study from the best tutors and mentors
and much more thanks once again
derivatives part 3 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 and
Martin now will take the example of
limit so these function is FX is going
to do and we have to find at 4 left hand
limit right hand limit any limit it
stalls Ram FX is going to always to you
do whatever you want but this gun is
always to this value will be something
like this this is the rather this is 1/2
stool this is 3 this is 4 this is 5 so
we have to find the left-hand limit at 4
right hand limit at 4 and the limit at
for example now if you see I mean to say
limit X 4 minus of X this is nothing but
left-hand limit so if you take any value
which is little less than 4 little less
than 4 pazam poof f of I'll take three
point nine nine nine nine
so f of three point nine nine but the
value do you put any other x you get to
only so f of three point nine days
similarly if you want to find the right
limit of this you will say limit X 4
plus of X that is equal to let's suppose
will take any whether that is greater
than phone and nearer to 4 there is 4
point 0 1 and this value is also because
you say any value of you see a given U
of X that is 2 this is the function you
take four point zero one you get two you
take three point nine nine nine you get
T's so here if you see the left is equal
to right correct the left limit is equal
to the right limit so we can say the
limit at X is equal to four exist since
the left is equal to right so we can say
that limit at X is equal to
they exist and the value of limit at X
is going to the forest by this since
this will be able to left in limit or
right limit in this case left right do
the same so this is this is what we have
done we have found the left limit and we
have found the right limit and we saw
that left and right there same that
means we can conclude that limit for
this function at X equal to 4 exists and
then we can say yeah this is the back we
take one more example if X is equal to X
square and then X is going to 4 we have
to find the left limit and right limit
let us draw this it's short once again
FX equal to X square we have to find the
limit left limit right limit and the
limit where X is almost equal to 4 so
let us draw the graph first let's say
this is 1 this is 2 this is 3 this is 4
5 6 so let's suppose the graph is
something like this X go to 0 so we have
equation y is equal to X square all
right X is going to 0 Y is equal to 0 X
is going to 1 Y is equal to 1 X equal to
2 y is equal to 2 squared - 4 X is equal
to 3 y is equal to 3 squares go to 9 so
let's draw this X going to 1 0 Y is
going to 0 X is equal to 1 Y is equal to
1 X is equal to 2 y is equal to 4 1 2
correct x equal to 3 y is equal to 9 so
my hair so we'll get a graph like this
actually so the graph will be something
like this it's it is the graph correct
so let's suppose this is 4 this is 4 X
good 2 3 y is equal to 9 that so this is
9 and X equal to 4 y is equal to 16
let's Appaloosa 16 this is the graph we
have similar go - house it will be
almost same it will be the mirror image
actually
mr. laugh now we have to find the left
limit and the right limit so let me put
a dot here at this point is equal to
four is this sorry this is extend is
wrap will extend this draw this is
sixteen actually this is three this is
nine okay now this is the wrap now this
is 16 X is going to 4 y is equal to 16
now the question is we have to find the
left limit left limit is you say limit X
is 4 - there is nothing where the value
of function that of FX is nothing but
value of function at 4 - it is three
point nine nine nine all right so what
is the value of function at 3.99 is
equal to three point nine nine nine
square all right say three point nine
nine square what you get is something
all fifteen point nine nine nine nine
this is where you get
similarly for right limit limit of X for
class of FX is nothing but f of four
point zero zero zero one correct that is
nothing but four point zero zero one
square because Y is equal to x squared
and that is nothing worth sixteen point
zero zero one six that is the value we
get almost that very we get so if you
say this fifteen point nine nine nine is
almost equal to 16 and 16 point zero
zero one is always almost equal to
sixteen so we can say that in this case
left limit is equal to right limit so
here you say left is equal to so we can
say that the limit also exists when I
say limit X tends to four we are not
used plus minus here of FX then that
value is so in this point at this point
four I am talking about left limit and
ever watching this
from this side I get value 16 when I
approve this point from this side right
side I get a low 16 so I say that limit
X equal to 4 or FX is nothing but 16 but
I have done here I have found the left
limit and the right limit and I observe
that left limit is equal to right limit
let's take one more example here we have
to find the limit for FX equal to X plus
1 where X is near to X limits to X is
almost equal to 2 so let's plot this
graph this is 1 this is 2 this is 3 this
is 4 and this is 0 so X go to 0 Y is
equal to 0 plus 1 1 X equal to 1 Y is
equal to 1 plus 1/2 it is going to do Y
is equal to 2 plus 1 3 X 3 y is what is
available so we get graph like this so
the graph to get something like this of
a straight line it should be a straight
line actually this is the graph again
now at X is going to to this point we
have to find the left limit and the
right limit so in case of left limit we
have to approve this point from this
side left side so when I say limit X
tends to 2 minus there is left limit of
f of X that is nothing but f of 1.999
let's of this value so what you get is
this is nothing but 1 point 9 9 9 9 plus
1 and this is nothing but 2 point 9 9 9
and that is almost equal to 3 so you see
that left limit of this function is 3
similarly now let me approach this
function from right side so when I say
limit of X of 2 class of FX that is
nothing but value of this function at 2
point 0 0 0 1 anywhere is near to 2 it
is greater than 2 so this is nothing but
2
zero zero zero zero one last one and
there is nothing but three point zero
zero zero one and this value is almost
equal to three so here you see the left
limit and the right limit both are equal
you see here left is equal to right and
that is equal to three so we can say
that in this case limit at X is equal to
4 FX exists limit for this exists and
that value is nothing but that is answer
very simple we have on the Left limit in
the right limit and we saw that with
left limit invite Lynda equal so we
throw on that left limit and right limit
are equal so limit for this function at
X equal to exists and that's value is 3
thank you visit exam viacom to watch
free education videos try free online
test
get the best quality study materials
study from the best tutors and mentors
and much more thanks once again
About this Video
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Nov 17, 2024 Last updated |
Video Description: Limits: Examples Part 1 for Commerce 2024 is part of Mathematics (Maths) Class 11 preparation.
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