show that the function f(x)=sin(cosx) , is continous
Answers
Step-by-step explanation:
Since, g(x) = sin x + cos x is a continuous function as it is forming with addition to two continuous functions , sin x and cos x. ... Hence, f(x) = |sin x + cos x| is a continuous function everywhere.
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
g(x) =
x⇒c
lim
sin(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
g(x) =
x⇒c
lim
sin(x) = sin(c)=g(c)
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
f(x) =
x⇒c
lim
cos(x) = cos(c)=f(c)
And this proves that cos(x) is continuous all across its domain
=> So by theorem, if function f and function g are continous, then f.g is also continous.
There for sin(x).cos(x) is continous