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CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsCONTINUITY AND DIFFERENTIABILITY3 Marks
Question
Discuss the continuity of the following functions:
$\text{f(x)} = \sin \text{x} . \cos \text{x}$
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Answer

Let a be an arbitrary real number then $^{\ \ \text{lim}}_{\text{x}\rightarrow\text{a}^{+}}\text{f(x)} \Rightarrow^{\ \ \text{lim}}_{\text{h}\rightarrow\text{0}}\text{f(a + h)} $
$\Rightarrow\ ^{\ \ \text{lim}}_{\text{h}\rightarrow\text{0}}\sin\text{(a + h)} . \cos (\text{a} + \text{h})$
$\Rightarrow\ ^{\ \ \text{lim}}_{\text{h}\rightarrow\text{0}}(\sin\text{a}\cos\text{ h} + \cos\text{a} \sin \text{h})(\cos \text{a}\cos\text{h}-\sin\text{a}\sin\text{h})$
$= (\sin \text{a}\cos0+\cos\text{a}\sin0) (\cos\text{a}\cos0 - \sin\text{a}\sin0)$
$=( \sin \text{a} + 0) ( \cos\text{a}-0)$
$= \sin \text{a} . \cos\text{a}= \text{f(a)}$
Similarly, we have $^{\ \ \text{lim}}_{\text{x}\rightarrow\text{a}^{-}}\text{f(x)} = \text{f(a)}$
$\therefore\ ^{\ \ \text{lim}}_{\text{x}\rightarrow\text{a}^{-}}\text{f(x)}= \text{f(a)}= ^{\ \ \text{lim}}_{\text{x}\rightarrow\text{a}^{+}}\text{f(x)}$
Therefore, f(x) is continuous at x = a.
Since, a is an arbitrary real number, therefore, $\text{f(x)}= \sin\text{x} . \cos\text{x}$ is continuous.

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