Question

It is given that $f^{\prime }\left(a\right)$ exists, then $\underset{x\to a}{lim}\frac{xf\left(a\right)-af\left(x\right)}{x-a}$ is equal to:

• A. $f\left(a\right)-a{f}^{\prime }\left(a\right)$
• B. $f^{\prime }\left(a\right)$
• C. $-f^{\prime }\left(a\right)$
• D. $f\left(a\right)+a{f}^{\prime }\left(a\right)$

Answer 1

Answer: (A)

$f\left(a\right)-a{f}^{\prime }\left(a\right)$

$L=\underset{x\to a}{lim}\frac{xf\left(a\right)-af\left(x\right)}{x-a}$

Since this is of the $\frac{0}{0}$ form, we can use L'Hospital's rule,

$L=\underset{x\to a}{lim}\frac{f\left(a\right)-\left\{a{f}^{\prime }\left(x\right)+f\left(x\right)\times 0\right\}}{1-0}$

$=\underset{x\to a}{lim}\frac{f\left(a\right)-a{f}^{\prime }\left(x\right)}{1}$

$=f\left(a\right)-a{f}^{\prime }\left(a\right)$