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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->0)` [(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) = `lim_(x->a^+)` f(x) = f(a)
⇒ f(x) is continuous at x = a.
∴ f(x) = sin x – cos x is everywhere continuous.
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