It is given that $f'(a)$ exists, then $\lim\limits_{x \rightarrow a} \frac{x f(a)-a f(x)}{x-a}$ is equal to
Answer & explanation
Correct answer: option 1
It is given that f'(a) exists.
∴ $\lim\limits_{x \rightarrow a} \frac{f(x)-f(a)}{x-a}=f'(a)$
Now,
$\lim\limits_{x \rightarrow a} \frac{x f(a)-a f(x)}{x-a}$
$=\lim\limits_{x \rightarrow a} \frac{(x-a) f(a)-a(f(x)-f(a))}{x-a}$
$=\lim\limits_{x \rightarrow a} f(a)-a\left\{\frac{f(x)-f(a)}{x-a}\right\}$
$=f(a)-a \lim\limits_{x \rightarrow a} \frac{f(x)-f(a)}{x-a}=f(a)-a f'(a)$