Discuss the continuity of the following function: f (x) = sin x × cos x - Mathematics

Question

Discuss the continuity of the following function:

f (x) = sin x × cos x

Sum

Solution

Let a be an arbitrary real number, then

f (a) = sina cosa

$lim_{x \to a^{+}} f (x) = lim_{h \to 0} [\sin (a + h) \cos (a + h)]$

= $lim_{h \to a^{+}} [(\sin a \cosh + \cos a \sinh) (\cos a \cosh - \sin a \sinh)]$

= sina cosa

$lim_{x \to a^{-}} f (x) = lim_{h \to 0} [\sin (a - h) \cos (a - h)]$

$= lim_{h \to 0} [(\sin a \cosh - \cos a \sinh) (\cos a \cosh + \sin a \sinh)]$

= sina cosa

∴ $lim_{x \to a^{-}} f (x) = lim_{x \to a^{+}} f (x) = f (a)$

= f (x) is continuous at x = a

So, f (x) = sinx. cosx is everywhere continuous.

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Algebra of Continuous Functions

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Chapter 5: Continuity and Differentiability - Exercise 5.1 [Page 160]

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Chapter 5 Continuity and Differentiability
Exercise 5.1 | Q 21 | Page 160

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