Question

${lim}_{x\to a}\frac{asinx-xsina}{a{x}^2-x{a}^2}$

Answer 1

${lim}_{x\to a}\frac{asinx-xsina}{a{x}^2-x{a}^2}={lim}_{x\to a}\frac{asinx-xsina}{ax(x-a)}$

Let t=x -a

Then, as $x\to a,t\to 0$

$\therefore {lim}_{x\to a}\frac{(asinx-xsina)}{ax(x-a)}={lim}_{t\to 0}\frac{asin(t+a)-(t+a)sina}{a(t+a)t}$

$={lim}_{t\to 0}\frac{asintcosa+asinacost-tsina-asina}{a(t+a)t}$

$={lim}_{t\to 0}\frac{asintcosa+asina(cost-a)-tsina}{a(t+a)t}={lim}_{t\to 0}\frac{asintcosa+asina\left(2si{n}^2\right(\frac{t}{2})-tsina}{a(t+a)t}$

$={lim}_{t\to 0}\frac{asintcosa}{a(t+a)t}+{lim}_{t\to 0}\frac{asina\left(2si{n}^2\right(\frac{t}{2})}{a(t+a)t}-{lim}_{t\to 0}\frac{tsina}{a(t+a)t}$

$=\frac{acosa}{a^2}+0\frac{sina}{a^2}=\frac{acosa-sina}{a^2}$