Show that sin x is continuous for every real x.
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Answers
Answered by
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Step-by-step explanation:
Let f(x)=sinx
Let c be any real number.
We know that A function is continuous at x=c
If L.H.L = R.H.L= f(c)
i.e.
x→c
lim
f(x)=
x→c
+
lim
f(x)=f(c)
Taking L.H.L
x→c
−
lim
f(x)
x→c
−
lim
(sinx)
since sin x is defined for every real number.
Putting x=c−h
x→c
−
c−h→x
−h→0
h→0
=
h→0
lim
sin(c−h)
=
h→0
lim
(sinccosh−sincsinh)
putting h = 0
=sinccos0−cosc.sin0
=sinc(1)−cosc.0
= sinc
Taking R.H.L
x→c
+
lim
f(x)
sinx→c
+
lim
sin(x)
putting x=c+h
h→0
lim
sin(c+h)
h→0
lim
sin(sinccosh+coscsinh)
putting h=0
=sinccos0+cosc.sin0
=sin(1)+cosc.0
=sinc
f(x)=sinx
f(c)=sinc
Hence L.H.L=R.H.L=f(c)
x→c
−
lim
f(x)=
x→c
+
lim
f(x)=f(c)
f(x) is continuous
so, is continous.
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