Question

If f(x) has a derivative at x=a, then ${lim}_{x\to a}\frac{xf\left(a\right)-af\left(x\right)}{x-a}$ is equal to

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

Answer 1

The correct option is A $f\left(a\right)-a{f}^{\prime }\left(a\right)$

$\underset{x\to }{lim}\frac{xf\left(a\right)-af\left(x\right)}{x-a}$
Applying L' Hospital Rule
$\Rightarrow \underset{x\to }{lim}\frac{f\left(a\right).1-a{f}^{\prime }\left(x\right)}{1}\Rightarrow f\left(a\right)-a{f}^{\prime }\left(a\right)$