lim_(x→a) ((asin x−xsin a)/(x−a))


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Question Number 200741 by Rydel last updated on 22/Nov/23

$\lim_{x \to a} \frac{a \sin x - x \sin a}{x - a}$
Commented by JDamian last updated on 22/Nov/23
really?
	Answered by JDamian last updated on 22/Nov/23

	$hint : x \sin a - a \sin a = (x - a) \sin a$
	Commented by Rydel last updated on 22/Nov/23

	$\lim_{x \to a} \frac{a \sin x - x \sin a}{x - a}$
	Answered by MM42 last updated on 22/Nov/23

	$h o p \to {l i m}_{x \to a} \frac{a c o s x - s i n a}{1}$ $= a c o s a - s i n a ✓$
	Answered by tri26112004 last updated on 23/Nov/23

	$= \lim_{x \to a} \frac{a (s i n x - s i n a) + a s i n a - x s i n a}{x - a}$ $= \lim_{x \to a} \frac{2 a c o s (\frac{x + a}{2}) s i n (\frac{x - a}{2})}{x - a} + \lim_{x \to a} \frac{(a - x) s i n a}{x - a}$ $= \lim_{x \to a} a c o s (\frac{x + a}{2}) - \lim_{x \to a} s i n a$ $= a c o s a - s i n a$
	Commented by Rydel last updated on 23/Nov/23

	$t h a n k y o u v e r y m u c h$
	Answered by MM42 last updated on 23/Nov/23

	$x - a = u$ $\Rightarrow {l i m}_{u \to 0} \frac{a s i n (u + a) - (u + a) s i n a}{u}$ $= {l i m}_{u \to 0} \frac{a s i n u c o s a + a s i n a c o s u - u s i n a - a s i n a}{u}$ $= {l i m}_{u \to 0} (\frac{s i n u}{u} a c o s a - s i n a - \frac{1 - c o s u}{u} a s i n a)$ $= a c o s a - s i n a$
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