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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) = sin a - cosa
`lim_(x->a ^+) f(x) = lim_(h->0) sin (a + h) - cos (a + h)`
= `lim_(h->0){(sina cosh + cosa sinh) - cosa cosh - 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->0)[(sina cosh - cosa sinh) - (cosa cosh + sina sinh)]`
= sin a - cosa
∴ `lim_(x->a^-) f(x) = lim_(x->a^+) f (x) = f(a)`
⇒ f(x) is continuous at x = a.
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