Introduction of Limit in Calculus - Limits & Derivatives Video Lecture - Class 11

some stuffs right suppose we have a
function f of X equals to X square minus
1 whole divided by X minus 1 right so
let me first draw the figure of this
function the functions figure will be a
little bit above like this right suppose
this is 1 this is 2 1 2 and this is the
origin right let me draw the line with
the different color
actually here only a gap you know now
let's see how to evaluate the functions
right how to get the value of this
function suppose if you put the value 1
in this function what you gonna do
what we'll get right simply 1 square
that is 1 1 minus 1 that's 0 right in
upside numerator you'll be getting 0 and
in Denver also 1 minus 1 will be getting
0 so 0 minus 0 divided by 0 will be
getting the infinity again right that's
undetermined value so what we are
getting here the function is not
existing over here right when the X is
equals to 1 when X is equals to 1 the
function is not existing here we are not
getting the value of x so as we were
going going towards this point suppose I
am giving the name suppose this point a
as we're approaching the value of this
point will be closing here right will be
going to close it closer to this point
and again as will be going closer and
closer from the right side also right
will be coming here right so to
calculate the value of this or the
height of this point a simply we take
now the limit as limit when X tends to 1
right X square minus 1 X minus 1 of this
function it will be exactly equals to
this to how you can simply do simple
just by taking some numbers from the
left side and from the right side also
just let me the rough over here suppose
all right this is X and this is f of X
suppose and now I'm taking the different
different numbers over here suppose
first I'm taking from right side
suppose I'm taking here 0.9 now if
output 0.9 in this function I'll be
getting 1.9 you can just pause this
video and calculate it into your
calculator you'll be getting 1.9 right
and again I'm going to put your zero
point nine nine right if I'll put the
value of x here 0.9 that means I'm going
closer here right I'm going to be closer
closer almost here so I'll be here
almost one output 0.99 I'll get the
value here again one point nine nine
right I'm going more closer to this 2
right here exactly it's 2 so I'm going
more closer to 2 not exactly yeah not
exactly what I'm taking you right so I'm
going more closer to this now I'm again
taking zero point nine nine nine nine
nine suppose four five times nine right
so again I will be coming more poser to
two nine nine nine nine nine right
that's three two right three two I'm
going more closer to nine right so as I
was approaching to near value of this
one from here near value of one as I was
taking near value of one more nearer I
was coming to the two right so the same
thing I'm doing now from the right side
also right now from here also I'm doing
the same thing
suppose I'm first doing here
suppose now I'm taking suppose one point
one so if I'll take the value of one
point one over here and if I'll put in
this function I'll get here two point
one right now I'm coming from this side
right one point one suppose I is here
one point one now I'm coming more closer
here so initially I was here almost now
I will be coming more closes since now
I'm taking one point suppose 0 1 so if I
will put the one point zero one in this
function I will get the value of two
point zero one right now I'm taking more
closer values suppose one point zero
zero zero
one so if I'll put 1.000 one over here
then again I will get to point zero zero
one again one there is one more day
right that three zero and then one right
so as I was taking the values from the
right side and coming closer to one
right coming closer to one right I was
getting the value more closer to this
tool right more closer to two to this
tool from the right side also and the
same from the left side also so but when
I was taking one there there we we do
not get any value that's undefined value
right that's undefined value for the one
right that undefined values so what we
can say now we are taking actually the
approaching fellow of this so we are
calculating the approaching value of
this so that's why we introduced a new
chapter that's limit over here to limit
taking the limit a limit of this X
square minus one a whole divided by X
minus 1 when X approaches 2 1 is equals
to 2 this is the limiting value or
approaching value of this function right
so we cannot calculate exactly the value
of this function in this right in this
graph so that's why we take the limiting
value right it is a good example of here
also you can see in this figure as we're
going closer and closer and closer
closer closer closer you can see here
the value the functions value you can
see here right right from see a number
is coming closer closer to 2 closer to 2
closer to 2 right closer to 2 more
closer more closer more closer right 6 7
see it's more closer and when we get the
value exactly 2 it's undefined see it's
undefined right when exactly we are
putting the 1 it were getting undefined
value exactly same when we are coming
closer from the right side again we're
getting closer to 2 closer to 2 you can
see the value of FX write f of X we're
coming more closer to 2 more closer to 2
more closer closer closer and closer but
when the exactly X is ax x equals to 1
we are getting the value on
find value right we are getting now on
different value when x equals to 1 we
are getting undefined value the same
thing right it's a very good right
actually I'm getting this from the
websites name that is called maths is
fun calm right you can check it out this
it's really wonderful right so I hope
you understand up a little bit
background a little bit concept of the
limit right so let me do some examples
here to make you clear in this server
suppose this is my graph now right and
it is a function suppose and it has a
fellow over here right suppose it is a
function f of X now so when we're coming
let me change the color a little more
better okay when we're coming closer and
closer closer to this point right as
we're coming approaching this fellow
from the left side also and we're
approaching from the right side all
right so what we can see the ending
point both of this in both side as we're
coming from the left side and right side
the any point is the same right so what
we can say the limit suppose this is
this value is L and this is X so when X
right suppose this x value is equals to
here when X tends to a right in this
graph suppose when X tends to a when we
come closer to X right so f of X have
the value equals to L right so when X
tends to a the f of X the value of the f
of X is almost equals to not exactly is
almost equals to L it's a positional
approaching well right so when X
approaches to a f of X also approaches
to L so like this but since we are going
from the left side also and the right
side also what we can say we have the
two limits right so this is called the
left-hand limits right and we generally
write like this limit when X tends to a
and
generally we take a negative sign here
right so limit f of X is equals to not
exactly equals right so L so this is
called the left hand limit right
it's called left hand limit left hand
limit now see here
so a minus we get we give the sign this
is the symbol we generally write the
left hand limit now see here also in
right side also suppose this color now
as I'm coming from the right side more
closer to this point right so again what
I can say we have the right hand limit
also and we generally represent it like
this limit X tends to a plus we give you
a positive sense because all the values
will be greater than e and here - that's
all we are putting here - because all
the values are less than the right so
this is a right hand limit now and f of
X is now a process to the value L right
so this is called the right hand limit
right hand limit and one limit any limit
only exists on the left hand limit and
the right hand limit have the same value
right if the right hand limit and the
left hand hibbett limit have the same
value then only that limit exists
otherwise it do not exist okay now let
me do another example over here suppose
now it is up
hmm okay
here is a function f of X suppose right
let me deal with the red color right you
know here here you're watching a jump
over here right it is a jump you know if
you come from the left side right you'll
get a value suppose this is Abe and this
is L suppose L 1 value you are getting
right when you are coming from the left
side so left hand limit what you can say
left hand limit when X approaches to a
minus f of X you are getting l1 value
suppose right and now when you're coming
from the right side right you are
getting something another value right
you're getting something another value
suppose it is l2 so our right hand limit
is when X approaches to a now plus f of
X we are getting we are approaching to a
value of l2 right so when you are coming
from the right side our right hand limit
is l2 and our left hand limit is l1 so
since L 1 is not equals to L 2 we can
see the limit in this graph does not
exist right limit does not exist so what
we can say now that to exist a limit the
left hand limit and the right hand limit
should must be equal halves it should be
the same value right it should have the
same value now I think you understand a
little bit concept over here now here is
a figure right
let's observe this figure one by one now
see here at the -2 point right at the -2
point we have the right hand limit right
but we do not have the left hand limit
so in this case what we can say sweet
okay here we have the right hand limit
but we didn't have the left hand limit
so that's what again we can say limit
does not exist
so from this point what we get that to
exist a limit we should have the
left-hand limit or the point the
limiting point should have we should get
the left-hand limit also and the right
hand limits also from the left hand also
and the from the right side also so if
there is not the common point means the
the same ending point from the left side
also and from the right side also then
we can say there is no existence of the
limit right now come here simply this is
why watching we're here here you are
watching here there from the left side
also we're getting the same value and
from the right side also we are getting
the same so left hand limit is equal to
the right hand limit so here limit
exists right and here also we are
getting the left hand limit equals to
the right hand limit that's why we can
say here also the limit exists but here
here is a jump you know again see here
pays a jump here is some value from the
left hand suppose l1 and from the right
side if you go it is l2
so since l1 l1 and l2 we're getting the
two different limit from the left hand
on the right side so here again we can
say limit do not exist right doesn't
exist it's not writing okay doesn't
exist limit doesn't exist here right so
I hope you got a little bit idea about
limit and in next videos we'll be doing
some little bit problems about the
limits and the continuity and then we'll
be doing lots of our about the
derivatives and we'll be starting
gradually we'll be starting the calculus
portion right so it is also the calculus
right so we'll meet in the next video
with a different the different example
of the limit okay well by uh limiting
next to a bike