Evaluate the following limits:
Question:

Evaluate the following limits:

$\lim _{h \rightarrow 0} \frac{(a+h)^{2} \sin (a+h)-a^{2} \sin a}{h}$

 

Solution:

$=\lim _{h \rightarrow 0} \frac{(a+h)^{2} \sin (a+h)-a^{2} \sin a}{h}$

$=\lim _{h \rightarrow 0} \frac{a^{2}(\sin (a+h)-\sin a)+2 a h \sin (a+h)+h^{2} \sin (a+h)}{h}$

$=2 a \sin a+0+\lim _{h \rightarrow 0} \frac{a^{2} \times 2 \times \cos \left(a+\frac{h}{2}\right) \times \sin h}{h}$

$=2 a \sin a+2 a^{2} \cos a$

$=2 a^{2} \cos a+2 a \sin a$

$\therefore \lim _{h \rightarrow 0} \frac{(a+h)^{2} \sin (a+h)-a^{2} \sin a}{h}=2 a^{2} \cos a+2 a \sin a$

 

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