Examples Limits of Trigonometric Functions Video Lecture | Mathematics for Grade 11

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derivatives part to earth is what you
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we have to find the value of sine 4x by
sine 2 X where X tends which in this
example we have to find the value of
sine is 4x by sine 2 X when X tends to 0
first thing what we do will put X equal
to 0 so you will get sine of hole in his
0 by sine of window 0 that becomes sine
0 by sine 0 that is 0 by this since it
is 0 by 0 form we have confident that we
can write this D write this equation to
some form there will be the ballot so
what we can do we can write this in this
form sine 4x by let's then multiply the
vector X and this sine 2 X we had so
there is a multiplying different places
all right that's what we have done here
right because we know that limit of sine
X by X where X tends to 0 is 1 this from
La Mina so we have just rewriting this
in this form sine 4x by 4x + sine 2 X 5
minutes let me talk extensively so what
we get here is this nothing but limit of
X tends to 0 sine of 4x violence in -
right limit X tends to 0 X by swing of 2
X into its X cancel when could scarcely
Utley - this is what we have got now we
know that limit of X tends to 0 sine is
1/2 for X is nothing but limit of 4 X 10
zeros into the first on the same thing
similarly limit of X tends to 0 sine X
by 2 s we can say 2 X 10 0 because X
tends to 0 to make us run the same thing
this value
is nothing but one because we have this
formula here X for X similarly this
value will also be one by one here also
X is equal to nothing but 2 X and this
is 1 so what you get is to ask them very
simple what we have done we have just
written this formula in this form sine X
by 4 X and here sine 2 X by 2 X and we
have use the formula limit of sine X by
X where extent is let's go to 1
extensive Elizabeth we won
so here what we have done we have made
4x x squared or 4x actually fit to the
formula and here x2 X you fit to the bar
this is a very simple pushing where we
have to write the value of tan X by X
where X 10 to 0 first we'll just put the
value of x 10 to 0 being income and 0 by
0 this becomes 0 by 0 so it is 0 by 0
that means we can rewrite this in this
red for tan x we can drive as sine X Phi
cos it's right by B writing this in
science by ptosis because it also will
be the same Gamla limit of sine X by X
where X tends to 0 is nothing but 1 so
we have this form what we can do we can
write in this form limit of X tends to 0
sine X by X into limit of X tends to 0 1
by correct so if you use this calm limit
of X tends to 0 sine x by X this becomes
1 and here X tends to 0 cos x kills us
put the value of course X go to 0 or 0
cos 0 is 1 we know this becomes 1 by 1
by 1 that is the design
see what we are doing here is we first
try to put the value of x go to zero in
most of the equations you get 0 by 0
form we know that we can somehow Rita is
the equation to make sure that the part
which is making a 0 by 0 is canceled so
here we have seen that
we have written this in sign mix by
Cossacks so we got signings bags form
and that Paulo is nothing but one so we
wrote this value as well but for one pi
cos x given X be 0 it is not a tonight
this one so we have put that Osborne is
0 and we got one by one to two letters
here we have to find the value of this
function extract as being C by C X
square plus BX plus a where X tends to 1
and a plus B plus these markets first
thing what we do it put the value of
expectable so what we get a into 1
square plus B into 1 plus C because we
have to fix going to 1 here similarly C
into 1 square plus B into 1 plus that is
nothing but a plus B plus C by C + b + a
or we can say this is nothing but a plus
B plus C a versus that is 1 since it is
1 it is a finite value we do not do
anything is that had this been 0 as
uniform we would have thought of
rewriting this but since the value is
already 1 we not do anything in this
thing is nice and it is already saying
that a plus B plus C is not equal to
where is it
and also we have to find the value of
this function X tends to 0 first thing
we put the value of x conducive so this
becomes cos sex 0 minus for this room
this is then they work in 5 minus
infinity so this is not incorrect form
so we have to rewrite this so let us do
that so we'll rewrite this cost X as
nothing but 1 by sine X you can write
and court as nothing but cos x minus
sine X why we did this you just wanted
to be ideas and we'll explore what is
the best possible way now we can say
that this is nothing but 1 minus cos x
by sine X correct this is nothing well
exchange usual one winds or subscribe
please the only form that we know that
sine X by X type so what we can do is we
can divide this whole thing by X because
we know sine X by X so what we'll do
we'll divide this by X and divide these
medals to say all right so what we can
write this as limit of X tends to 0 1
minus of sex by X divided by limit of X
tends to 0 sine X types correct and this
value we know is 1 now we have to find
the value of this 1 minus cos x by X
again we can write this change for
expectant because we know sine X by X
let me text it we do but we do not know
cos x by its limit until that whether we
don't know should I write in this map so
this becomes limit of X is to 0 means
for x by 2 by X divided by 1 the way
better this is much too sine square X by
2 what we can do we can put X there so
what will get this enhanced limit of X
tends to 0 we get sine X by 2 sine X by
2 by but we have just taken this 2 here
now you can rewrite this as nothing
we're limit of sine X by 2 X is 2 0 into
limit of sine X by 2 pi you can say
anyway principle because X tends to 0 or
X by 2 then 0 is 1 the same thing all
right if didn't this now let's put the
ban explanation will become sine of 0 by
2 n this becomes 1 because this is the
standard form that we have at this sine
of X
X where X tends to 0 this becomes 1 this
compliance so you found maybe that light
here is becomes smooth becomes 1 and
sine 0 by 2 in sine 0 that is 0 to 1 the
proper value of this conscience is if
that designs it is simple what we have
that we have first put the value of x
because 0 and responsibly got infinite
so what we have done again we have
clearly done this equation in this form
and 1 minus cos x again we use two sine
square x pi/2 because we know the
formula really sentence works
oh and we have found the incidence in an
example we have to find the value of tan
2 X by X minus Phi by 2 living extension
since we know only emit X tends to 0 or
the formula which we have because
looking at the former we can say that we
can somehow convert this into sine X by
X form all right because tan X will
design is very cos x and we know only of
X tends to 0 form so what we can do here
you can see that means more Y is equal
to let suppose X minus L it as you
listen so when I say that X is tending
by way to you you can say that X minus
PI by 2 is tending to PI by 2 minus Phi
by 2 let us look and I expands by
writing with you advise against a by
tension so we can rewrite this equation
as tan off X is nothing but y plus PI
over 2 not it is y plus a better correct
by X minus PI by 2 is nothing but Y my
initial of X tends to PI by 2 we can say
that by James is room correct so we have
read it in this form so why because we
have all this sine X by X where X tends
to 0 this form okay so we have read it
in this in this form this becomes
nothing but tan - why
and by plus pi X by Y by text so tan of
X the size on this n of X actually so
this becomes tan of 2 y by y here Y
tends to limit already this is the very
very but this looks that begins simple
find the back so write this is nothing
but tan of 2 I can write sign up to y by
cos of 2 when y because I know the value
of sine of X by X by 1 by y y tends to
so all right now we have the bomb we
know the formula of sine 2 pi by 2 y
because sine of X by X so sine 2 pi by 2
y so I divide in x so this efficient I
can say then this is nothing the limit
of white and streets lead over sine of Y
by Y right into by pass to you I write
this is a limit of my 10 students
correct
so I say limit of identity or I can also
limit up to identity one of the same
thing and this equation if you see is
almost similar to this equation X tends
to 0 where sine X by X here X is nothing
but 2 y so I can write this value as 1
this becomes 1 into 2 by cos of equal 0
that is cos of 0 and that cos of 0 is 1
so this becomes money that is answer is
that very simply what we have done as we
have certain that this was 10 X by X
form and it was X minus extension PI by
2 but we know that always we have this
formula of sliding by X when X tends to
0 so we assume that Y is equal to X
minus PI by 2 so we say X is tending to
PI by 2 again say X minus PI by 2 KT to
0 now we can say Y is tending to 0 so we
really devote this equation in this form
and to y 2 y minus PI by 2
by plus PI by 2 pi by peer by 10 to 0 or
this is nothing but 10 10 to Y by Y by Y
tends to 0 then off to Y we can see is
nothing about sine of 2 y by cos of 2 1
and then we have used formula that limit
of sine Y by Y where Y tends to do is
nothing but 1 and the invert them answer
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