Discuss the continuity of sine function. - Vidyadip

Question

Discuss the continuity of sine function.

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Answer

To see this we use the following facts
$\mathop {\lim }\limits_{x \to 0} $ sin x = 0 = sin 0. So, sin x is continuous at x = 0.
Now, observe that f(x) = sin x is defined for every real number. Let c be a real number. Put x = c + h. If x $\rightarrow$ c then, h $\rightarrow$ 0. Therefore,
$\mathop {\lim }\limits_{x \to c} f(x) = \mathop {\lim }\limits_{x \to c} $ sin x
= $\mathop {\lim }\limits_{h \to 0} $ sin (c + h)
= $\mathop {\lim }\limits_{h \to 0} $ [sin c cos h + cos c sin h]
= $\mathop {\lim }\limits_{h \to 0} $ [sin c cos h] + $\mathop {\lim }\limits_{h \to 0} $[cos c sin h]
= sin c + 0 = sin c = f (c)
Thus $\mathop {\lim }\limits_{x \to c} f(x)$ = f(c) and hence f is a continuous function.

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