Q. If $f(x)$ has a derivative at $x = a,then\lim_{x \rightarrow a}\frac{xf(a) - af(x)}{x - a}$ is equal to
A) $f(a) - af^{'}(a)$
B) $af(a) - f^{'}(a)$
C) $f(a) + f^{'}(a)$
D) $af(a) + f^{'}(a)$
Explanation
(a)
$\lim_{x \rightarrow a}\frac{xf(a) - af(x)}{x - a}$
$= \lim_{x \rightarrow a}\frac{xf(a) - af(a) - af(x) + af(a)}{x - a}$
$= \lim_{x \rightarrow a}\frac{f(a)(x - a) - a\lbrack f(x) - f(a)\rbrack}{x - a}$
$= \lim_{x \rightarrow a}{\frac{f(a)(x - a)}{x - a} -}a\lim_{x \rightarrow a}\frac{f(x) - f(a)}{(x - a)}$
$= f(a) - af^{'}(a)$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.