Question

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

Answer 1

We have,

${lim}_{x\to a}\frac{x\sin a-a\sin x}{x-a}$

Now,

${lim}_{x\to a}\frac{x\sin a-a\sin x}{x-a}\frac{0}{0}form$

Using L’ Hospital rule and we get,

${lim}_{x\to a}\frac{x\frac{d}{dx}\sin a+\sin a\frac{d}{dx}x-a\frac{d}{dx}\sin x}{\frac{d}{dx}x-\frac{d}{dx}a}$

$\Rightarrow {lim}_{x\to a}\frac{x\times 0+\sin a\times 1-a\cos x}{1-0}$

$\Rightarrow {lim}_{x\to a}\sin a-a\mathrm{cosx}$

Taking limit and we get,

$\Rightarrow \sin a-a\cos a$

Hence, this is the answer.