if cos( x/2).cos(x/22).cos(x/23). = Sinx/x,
Answers
Step-by-step explanation:
Here clearly both sinx and cosx are defined in their domain.
Let's assume that
g(x)=
sinx
and
f(x)=
cosx
=> Let's first prove that g(x) is continuous in it's domain.
Let c be a real number, put
x=
c+
h
So if
x⇒
c
then it means that
h⇒
0
x⇒c
lim
sin(x)
x⇒c
lim
g(x) =
Put
x=
h+
c
And as mentioned above, when
x⇒
c
then it means that
h⇒
0
Which gives us
h⇒0
lim
sin(c+
h)
Expanding
sin(c+
h)
=
sin(h)cos(c)+
cos(h)sin(c)
Which gives us
h⇒0
lim
sin(h)cos(c)+
cos(h)sin(c)
=
sin(c)cos(0)+
cos(c)sin(0)
=
sin(c)
So here we get,
x⇒c
lim
sin(x)
=
sin(c)=
g(c)
x→c
lim
g(x) =
And this proves that sin(x) is continuous all across its domain
=> Let's prove that
f(x)=
cos(x)
is continuous in it's domain.
Let c be a real number, put
x=
c+
h
So if
x⇒
c
then it means that
h⇒
0
f(c)= cos(c)
x⇒c
lim
f(x) =
x⇒c
lim
cos(x)
Put
x=
h+
c
And as mentioned above, when
x⇒
c
then it means that
h⇒
0
Which gives us
h⇒0
lim
cos(c+
h)
Expanding
cos(h+
c)
=
cos(h)cos(c)−
sin(h)sin(c)
Which gives us
h⇒0
lim
cos(h)cos(c)−
sin(h)sin(c)
=
cos(c)cos(0)−
sin(c)sin(0)
=
cos(c)
This gives us
x⇒c
lim
cos(x)
=
cos(c)=
f(c)
x⇒c
lim
f(x) =
And this proves that cos(x) is continuous all across its domain
=> So by theorem, if functionf
and functiongare continous, then
f.g
is also continous.
There for sin(x).cos(x) is continous.
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