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Question
\[\lim_{h \to 0} \frac{\left( a + h \right)^2 \sin \left( a + h \right) - a^2 \sin a}{h}\]
Solution
\[\lim_{h \to 0} \left[ \frac{\left( a + h \right)^2 \sin \left( a + h \right) - a^2 \sin a}{h} \right]\]
\[= \lim_{h \to 0} \left[ \frac{a^2 \sin \left( a + h \right) + h^2 \sin \left( a + h \right) + 2ah \sin\left( a + h \right) - a^2 \sin a}{h} \right]\]
\[ = \lim_{h \to 0} \left[ a^2 \left\{ \frac{\sin\left( a + h \right) - \sin\left( a \right)}{h} \right\} + \frac{h^2 \sin\left( a + h \right)}{h} + \frac{2ah \sin\left( a + h \right)}{h} \right]\]
\[\text{ Dividing and multiplying the denominator by } 2:\]
\[ \lim_{h \to 0} \left[ a^2 \left\{ \frac{2\cos\left( \frac{a + h + a}{2} \right) \sin\left( \frac{a + h - a}{2} \right)}{2 \times \frac{h}{2}} \right\} + h\sin\left( a + h \right) + 2a \sin\left( a + h \right) \right]\]
\[ = \lim_{h \to 0} \left[ a^2 \cos \left( \frac{a + h + a}{2} \right) + h \sin \left( a + h \right) + 2a \sin \left( a + h \right) \right]\]
\[ = a^2 \cos a + 0 + 2a \sin a\]
\[ = a^2 \cos a + 2a \sin a\]
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