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evaluate lim h tends to 0 [ (a+h)^2 sin (a+h) - a^2sin a] / h

$\lim_{h \to 0} \frac{{(a + h)}^{2} \sin (a + h) - a^{2} \sin a}{h} = \lim_{h \to 0} \frac{a^{2} \sin (a + h) + h^{2} \sin (a + h) + 2 a h \sin (a + h) - a^{2} \sin a}{h} = \lim_{h \to 0} \frac{a^{2} \sin (a + h) - a^{2} \sin a + h^{2} \sin (a + h) + 2 a h \sin (a + h)}{h} = \lim_{h \to 0} \frac{a^{2} [\sin (a + h) - \sin a] + h^{2} \sin (a + h) + 2 a h \sin (a + h)}{h} = \lim_{h \to 0} \frac{a^{2} [\sin (a + h) - \sin a]}{h} + \lim_{h \to 0} \frac{h^{2} \sin (a + h)}{h} + \lim_{h \to 0} \frac{2 a h \sin (a + h)}{h} = \lim_{h \to 0} \frac{a^{2} [2 \cos (\frac{a + h + h}{2}) \sin (\frac{a + h - h}{2})]}{h} + \lim_{h \to 0} h \sin (a + h) + \lim_{h \to 0} 2 a \sin (a + h) = \lim_{h \to 0} \frac{a^{2} [2 \cos (\frac{a + 2 h}{2}) \sin (\frac{a}{2})]}{h} + \lim_{h \to 0} h \sin (a + h) + \lim_{h \to 0} 2 a \sin (a + h) = \lim_{h \to 0} \frac{2 \cos (\frac{a + 2 h}{2})}{(\frac{a + 2 h}{2})} \times (\frac{a + 2 h}{2}) \times a^{2} \sin (\frac{a}{2}) \times \frac{1}{h} + \lim_{h \to 0} h \sin (a + h) + \lim_{h \to 0} 2 a \sin (a + h) = \lim_{h \to 0} \frac{2 \cos (\frac{a + 2 h}{2})}{(\frac{a + 2 h}{2})} \times (\frac{a + 2 h}{2 h}) \times a^{2} \sin (\frac{a}{2}) + \lim_{h \to 0} h \sin (a + h) + \lim_{h \to 0} 2 a \sin (a + h) = 2 \times (0 + 1) \times a^{2} \sin (\frac{a}{2}) + 0 + 2 a \sin (a + 0) = 2 a^{2} \sin (\frac{a}{2}) + 2 a \sin a$

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