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Evaluate the Following Limit: Lim H → 0 ( a + H ) 2 Sin ( a + H ) − a 2 Sin a H - Mathematics

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

\[\lim_{h \to 0} \frac{\left( a + h \right)^2 \sin\left( a + h \right) - a^2 \sin a}{h}\]
\[ = \lim_{h \to 0} \frac{\left( a^2 + 2ah + h^2 \right)\sin\left( a + h \right) - a^2 \sin a}{h}\]
\[ = \lim_{h \to 0} \frac{\left( 2ah + h^2 \right)\sin\left( a + h \right) + a^2 \sin\left( a + h \right) - a^2 \sin a}{h}\]
\[ = \lim_{h \to 0} \frac{\left( 2ah + h^2 \right)\sin\left( a + h \right)}{h} + \lim_{h \to 0} \frac{a^2 \sin\left( a + h \right) - a^2 \sin a}{h}\]
\[ = \lim_{h \to 0} \left( 2a + h \right)\sin\left( a + h \right) + a^2 \lim_{h \to 0} \frac{\sin\left( a + h \right) - \sin a}{h}\]
\[\lim_{h \to 0} \frac{\left( a + h \right)^2 \sin\left( a + h \right) - a^2 \sin a}{h}\]
\[ = \lim_{h \to 0} \frac{\left( a^2 + 2ah + h^2 \right)\sin\left( a + h \right) - a^2 \sin a}{h}\]
\[ = \lim_{h \to 0} \frac{\left( 2ah + h^2 \right)\sin\left( a + h \right) + a^2 \sin\left( a + h \right) - a^2 \sin a}{h}\]
\[ = \lim_{h \to 0} \frac{\left( 2ah + h^2 \right)\sin\left( a + h \right)}{h} + \lim_{h \to 0} \frac{a^2 \sin\left( a + h \right) - a^2 \sin a}{h}\]
\[ = \lim_{h \to 0} \left( 2a + h \right)\sin\left( a + h \right) + a^2 \lim_{h \to 0} \frac{\sin\left( a + h \right) - \sin a}{h}\]

 

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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 29 Limits
Exercise 29.7 | Q 62 | Page 51
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