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lim_x-0(a-x)²sin(a+x) -a²sina / x

$\lim_{x \to 0} \frac{(a - x)^{2} \sin (a + x) - a^{2} \sin a}{x} = \lim_{x \to 0} \frac{(a^{2} - 2 a x + x^{2}) \sin (a + x) - a^{2} \sin a}{x} = \lim_{x \to 0} \frac{a^{2} \sin (a + x) - 2 a x \sin (a + x) + x^{2} \sin (a + x) - a^{2} \sin a}{x} = \lim_{x \to 0} \frac{a^{2} [\sin (a + x) - \sin a] + (x^{2} - 2 a x) \sin (a + x)}{x} = \lim_{x \to 0} \frac{a^{2} [2 \sin (\frac{a + x - a}{2}) \cdot \cos (\frac{a + x + a}{2})] + (x^{2} - 2 a x) \sin (a + x)}{x} = \lim_{x \to 0} \frac{a^{2} [2 \sin x \cdot \cos (\frac{2 a + x}{2})] + (x^{2} - 2 a x) \sin (a + x)}{x} = \lim_{x \to 0} 2 a^{2} (\frac{\sin x}{2}) \cdot \cos (\frac{2 a + x}{2}) + \lim_{x \to 0} \frac{x (x - 2 a) s i n (a + x)}{x} = 2 a^{2} \cdot \cos (a) + \lim_{x \to 0} (x - 2 a) \sin (a + x) = 2 a^{2} \cos a - 2 a \sin a = 2 a (a \cos a - \sin a)$

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