Evaluate : $\lim\limits_{h\to 0}\large\frac{(a+h)^2\sin(a+h)-a^2\sin a}{h}$

Question

$\begin{array}{1 1}(a)\;a^2\cos a+2a\sin a&(b)\;a^2\cos a-2a\sin a\\(c)\;a\cos a+a^2\sin a&(d)\;a\cos a+a^2\sin a\end{array}$

Answer 1

$\lim\limits_{h\to 0}\large\frac{(a+h)^2\sin(a+h)-a^2\sin a}{h}$

$\Rightarrow \lim\limits_{h\to 0}\large\frac{a^2[\sin(a+h)-\sin a]+2ah\sin(a+h)+h^2\sin(a+h)}{h}$

$\Rightarrow \lim\limits_{h\to 0}\large\frac{a^2[2\cos(a+h/2)\sin h/2]}{2\times h/2}+$$2a\sin(a+h)+h\sin(a+h)$

$\Rightarrow a^2\cos a+2a\sin a$

Hence (a) is the correct answer.