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  • Examine the continuity of f, where f is defined by f x = sin x-cos x for x not equal to 0 -1 for x= 0

25. Examine the continuity of f, where f is defined by 

f (x) = \left\{\begin{matrix} \sin x - \cos x & if x \neq 0 \\ -1 & if x = 0 \end{matrix}\right.

Answers (1)

best_answer

Given function is
f (x) = \sin x - \cos x
Given function is defined for all real number
We, know that if two function g(x) and h(x) are continuous then g(x)+h(x) , g(x)-h(x) , g(x).h(x) allare continuous
Lets take g(x) = sin x   and    h(x) = cos x
Let suppose  x = c + h
if  x \rightarrow c , \ then \ h \rightarrow 0
g(c) = \sin c\\ \lim_{x\rightarrow c}g(x) = \lim_{x\rightarrow c}\sin x = \lim_{h\rightarrow 0}\sin (c+h)\\ We \ know \ that\\ \sin(a+b) = \sin a \cos b + \cos a\sin b\\ \lim_{h\rightarrow 0}\sin (c+h) = \lim_{h\rightarrow 0}(\sin c\cos h + \cos c \sin h) = \lim_{h\rightarrow 0}\sin c\cos h + \lim_{h\rightarrow 0}\cos c \sin h
                                                                                                =\sin c\cos 0 + \cos c \sin 0 = \sin c
\lim_{x\rightarrow c}g(x) = g(c)
Hence, function g(x) = \sin x is a continuous function
Now,
h(x) = cos x
Let suppose  x = c + h
if  x \rightarrow c , \ then \ h \rightarrow 0
h(c) = \cos c\\ \lim_{x\rightarrow c}h(x) = \lim_{x\rightarrow c}\cos x = \lim_{h\rightarrow 0}\cos (c+h)\\ We \ know \ that\\ \cos(a+b) = \cos a \cos b + \sin a\sin b\\ \lim_{h\rightarrow 0}\cos (c+h) = \lim_{h\rightarrow 0}(\cos c\cos h + \sin c \sin h) = \lim_{h\rightarrow 0}\cos c\cos h + \lim_{h\rightarrow 0}\sin c \sin h
                                                                                                 =\cos c\cos 0 + \sin c \sin 0 = \cos c
\lim_{x\rightarrow c}h(x) = h(c)
Hence, function h(x) = \cos x is a continuous function
We proved independently that sin x and cos x is continuous function
So, we can say that
f(x) = g(x) - h(x) = sin x - cos x is also a continuous function

When x = 0
f (0) = -1\\ \lim_{x\rightarrow 0^-}f(x) = \sin 0 - \cos 0 = -1\\ \lim_{x\rightarrow 0^+}f(x) = \sin 0 - \cos 0 = -1 \\ R.H.L. = L.H.L. = f(0)
Hence, function is also continuous at x = 0

Posted by

Gautam harsolia

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