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प्रश्न
Discuss the continuity of the following function:
f (x) = sin x × cos x
उत्तर
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->a^+)[(sina cosh + cosa sinh) (cosa cosh - sina sinh)]`
= sina cosa
`lim_(x->a^-)f(x) = lim_(h->0)[sin(a - h) cos (a - h)]`
`= lim_(h->0) [(sin a cosh - cosa sinh) (cosa cosh + sina sinh)]`
= sina cosa
∴ `lim_(x->a^-) f(x) = lim_(x->a^+) f(x) = f (a)`
= f (x) is continuous at x = a
So, f (x) = sinx. cosx is everywhere continuous.
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