1.

Discuss the continuity of the following functions:(a) `f(x) = s in x + cos x` (b) `f(x) = s in x cos x` (c) `f(x) = s in x dot cos x`

Answer» (a) sin x and cos x are always continuous functions.
`:.` sin x + cos x is also a continuous function.
(b) `f(x) - sin x - cos x`
`= sqrt(2)((1)/(sqrt(2))sin x (1)/sqrt(2)cosx )`
`= sqrt(2)("cos"(pi)/(4) sin x - "sin" (pi)/(4) cosx)`
`= sqrt(2)sin(x -(pi)/(4) )`
we know that sine function is always continuous.
`:. sin (x - (pi)/(4)` is continuous function.
`rArr sqrt (2)"sin" (x - (pi)/(4))` is continuous function.
`rArr f(x) `is continuous function
(c) `f(x) = sin x cos x`
`=(1)/(2).2 "sin x cos x"= (1)/(2)" sin " 2x`
we know that sine function is always continuous.
`:. sin 2x` is a continuous.
`rArr (1)/(2) "sin "2x` is a continuous function.
`rArr f(x) ` is a continuous functions.


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