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Question
Discuss the continuity of the following function:
f(x) = sin x + cos x
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Solution
Let a be an arbitrary real number.
∴ f(a) = sin a + cos a
`lim_(x->a^+)` f(x) = `lim_(h->0)` [sin (a + h) + cos (a + h)]
= `lim_(h->0)` [(sin a cos h + cos a sin h) + (cos a cos h − sin a sin h)]
= sin a cos 0 + cos a sin 0 + cos a cos 0 − sin a sin 0
= sin a (1) + cos a (0) + cos a (1) − sin a (0)
= sin a + cos a
`lim_(x->a^-)` f(x) = `lim_(h->0)` [sin (a − h) + cos (a − h)]
= `lim_(h->a^-)` [(sin a cos h - cos a sin h) + (cos a cos h + sin a sin h)]
= sin a + cos a
∴ `lim_(x->a^-)` f(x) = f(a) = `lim_(x->a^+)` f(x)
⇒ f(x) is continuous at x = a.
∴ f(x) = sin x + cos x is everywhere continuous.
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