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Question
Examine the continuity of the following:
x + sin x
Solution
Let f(x) = sin x
f(x) is defined at all points of R.
Let x0 be an arbitrary point in R.
`lim_(x -> x_0) f(x) = lim_(x -> x_0) (x + sin x)`
= xo + sin x0 ........(1)
f(xo) = xo + sin xo ........(2)
From equations (1) and (2) we get
`lim_(x -> x_0) f(x) = f(x_0)`
∴ At all points of R, the limit of f(x) exists and is equal to the value of the function.
Thus, f(x) satisfies ail conditions for continuity.
Therefore, f(x) is continuous at all points of f(x).
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