Q. It is given that $f^{'}(a)$ exists, then $\lim_{x \rightarrow a}\frac{xf(a) - af(x)}{x - a}$ is equal to
A) $f(a) - a\ f^{'}(a)$
B) $f^{'}(a)$
C) $- f^{'}(a)$
D) $f(a) + a\ f^{'}(a)$
Explanation
(a)
Since, $f^{'}(a)$ exists.
$\therefore\ \ f^{'}(a) = \lim_{h \rightarrow 0}\frac{\ f(x) - \ f(a)}{x - a}$
Now, $\lim_{x \rightarrow a}\frac{xf(a) - af(x)}{x - a}$
$= \lim_{x \rightarrow a}\frac{xf(a) - af(a) + af(a) - af(x)}{x - a}$
$= \lim_{x \rightarrow a}\frac{f(a) - (x - a)}{(x - a)} - a\ \lim_{x \rightarrow a}\frac{\ f(x) - \ f(a)}{(x - a)}$
$= f(a) - a\ f^{'}(a)$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.