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Question
Discuss the continuity of the following function:
f(x) = sin x + cos x
Solution
Let a be an arbitrary real number then f(a) = sina + cosa
`lim_(x->a^+) f(x) = lim_(h->0)[sin(a + h) + cos(a+h)]`
`lim_(h->0){(sin acosh + cosa sin h) + (cos a cos h - sina sinh)}`
= sina cos0 + cosa sin0 + cosa cos0 - sina sin0
= sina (1) + cosa(0) + cosa(1) - sina(0)
= sina + cosa
`lim_(x->a^-)f(x) = lim_(h->0)[sin(a-h) + cos (a - h)]`
= `lim_(h->a^-)[(sina cosh - cosa sinh) + (cosa cosh + sina sinh)]`
= sina + cosa
∴ `lim_(x->a^-) f(x) = f (a) = lim_(x->a^+) f(x)`
= f(x) is continuous at x = a
∴ f(x) = sinx + cosx is everywhere continuous.
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