lim_(x→a) ((x^n sin a−a^n sin x)/(x−a))


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Question Number 200840 by Rydel last updated on 24/Nov/23

$\lim_{x \to a} \frac{x^{n} \sin a - a^{n} \sin x}{x - a}$
	Answered by MM42 last updated on 24/Nov/23

	${l i m}_{x \to a} \frac{x^{n} s i n a - a^{n} s i n a + a^{n} s i n a - a^{n} s i n x}{x - a}$ $= {l i m}_{x \to a} \frac{(x^{n} - a^{n}) s i n a - (s i n x - s i n a) a^{n}}{x - a}$ $= {l i m}_{x \to a} (\frac{(x - a) (x^{n - 1} + x^{n - 1} a + . . . + a^{n - 1}) s i n a}{x - a} - \frac{2 s i n (\frac{x - a}{2})}{x - a} \times c o s (\frac{x + a}{2}) \times a^{n})$ $= {n a}^{n - 1} s i n a - c o s a ✓$
	Commented by Rydel last updated on 24/Nov/23

	$t h a n k y o u v e r y m u c h$
	Answered by BaliramKumar last updated on 24/Nov/23

	$\lim_{x \to a} \frac{\frac{d}{dx} (x^{n} \sin a - a^{n} \sin x)}{\frac{d}{dx} (x - a)} = \frac{n x^{n - 1} s i n a - a^{n} c o s x}{1}$ ${n a}^{n - 1} s i n a - a^{n} c o s a = \frac{a^{n} (n s i n a - a c o s a)}{a}$
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