Question

It is given that $f'(a)$ exists, then $\displaystyle\lim _{x \rightarrow a} \frac{x f(a)-a f(x)}{(x-a)}$ is equal to :

• A. $f(a)-a f'(a)$
• B. $f' (a)$
• C. $-f' (a)$
• D. $f(a)+af'(a)$

Answer 1

Answer: (A)

By definition $f'(a)=\displaystyle\lim _{x \rightarrow a} \frac{f(x)-f(a)}{x-a}$,
Given $\displaystyle\lim _{x \rightarrow a} \frac{x f(a)-a f(x)}{x-a}$
$=\displaystyle\lim _{x \rightarrow a} \frac{x f(a)-x f(x)+x f(x)-a f(x)}{(x-a)}$
$=\displaystyle\lim _{x \rightarrow a}\left[\frac{x[f(a)-f(x)]}{(x-a)}+\frac{f(x)(x-a)}{(x-a)}\right]$
$=\displaystyle\lim _{x \rightarrow a}-\frac{x[f(x)-f(a)]}{x-a}+\lim _{x \rightarrow a} f(x)$
$=-a f'(a)+f(a)=f(a)-a f'(a)$