Evaluate the following limit:
\[\lim_{h \to 0} \frac{\left( a + h \right)^2 \sin\left( a + h \right) - a^2 \sin a}{h}\]
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Solution
\[\lim_{h \to 0} \frac{\left( a + h \right)^2 \sin\left( a + h \right) - a^2 \sin a}{h}\]
\[ = \lim_{h \to 0} \frac{\left( a^2 + 2ah + h^2 \right)\sin\left( a + h \right) - a^2 \sin a}{h}\]
\[ = \lim_{h \to 0} \frac{\left( 2ah + h^2 \right)\sin\left( a + h \right) + a^2 \sin\left( a + h \right) - a^2 \sin a}{h}\]
\[ = \lim_{h \to 0} \frac{\left( 2ah + h^2 \right)\sin\left( a + h \right)}{h} + \lim_{h \to 0} \frac{a^2 \sin\left( a + h \right) - a^2 \sin a}{h}\]
\[ = \lim_{h \to 0} \left( 2a + h \right)\sin\left( a + h \right) + a^2 \lim_{h \to 0} \frac{\sin\left( a + h \right) - \sin a}{h}\]
\[\lim_{h \to 0} \frac{\left( a + h \right)^2 \sin\left( a + h \right) - a^2 \sin a}{h}\]
\[ = \lim_{h \to 0} \frac{\left( a^2 + 2ah + h^2 \right)\sin\left( a + h \right) - a^2 \sin a}{h}\]
\[ = \lim_{h \to 0} \frac{\left( 2ah + h^2 \right)\sin\left( a + h \right) + a^2 \sin\left( a + h \right) - a^2 \sin a}{h}\]
\[ = \lim_{h \to 0} \frac{\left( 2ah + h^2 \right)\sin\left( a + h \right)}{h} + \lim_{h \to 0} \frac{a^2 \sin\left( a + h \right) - a^2 \sin a}{h}\]
\[ = \lim_{h \to 0} \left( 2a + h \right)\sin\left( a + h \right) + a^2 \lim_{h \to 0} \frac{\sin\left( a + h \right) - \sin a}{h}\]
\[ = \lim_{h \to 0} \frac{\left( a^2 + 2ah + h^2 \right)\sin\left( a + h \right) - a^2 \sin a}{h}\]
\[ = \lim_{h \to 0} \frac{\left( 2ah + h^2 \right)\sin\left( a + h \right) + a^2 \sin\left( a + h \right) - a^2 \sin a}{h}\]
\[ = \lim_{h \to 0} \frac{\left( 2ah + h^2 \right)\sin\left( a + h \right)}{h} + \lim_{h \to 0} \frac{a^2 \sin\left( a + h \right) - a^2 \sin a}{h}\]
\[ = \lim_{h \to 0} \left( 2a + h \right)\sin\left( a + h \right) + a^2 \lim_{h \to 0} \frac{\sin\left( a + h \right) - \sin a}{h}\]
\[\lim_{h \to 0} \frac{\left( a + h \right)^2 \sin\left( a + h \right) - a^2 \sin a}{h}\]
\[ = \lim_{h \to 0} \frac{\left( a^2 + 2ah + h^2 \right)\sin\left( a + h \right) - a^2 \sin a}{h}\]
\[ = \lim_{h \to 0} \frac{\left( 2ah + h^2 \right)\sin\left( a + h \right) + a^2 \sin\left( a + h \right) - a^2 \sin a}{h}\]
\[ = \lim_{h \to 0} \frac{\left( 2ah + h^2 \right)\sin\left( a + h \right)}{h} + \lim_{h \to 0} \frac{a^2 \sin\left( a + h \right) - a^2 \sin a}{h}\]
\[ = \lim_{h \to 0} \left( 2a + h \right)\sin\left( a + h \right) + a^2 \lim_{h \to 0} \frac{\sin\left( a + h \right) - \sin a}{h}\]
Is there an error in this question or solution?
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