Question

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

Given:-

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

$\Rightarrow$ As its $\left(\%\right)$ form so we apply L.Hospital Rule

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

$\Rightarrow$ Now we apply limit $x\to a$

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