Question:
Let $f(x)$ be a differentiable function at $x=a$ with $f^{\prime}(a)=2$ and $f(a)=4$. Then $\lim _{x \rightarrow a} \frac{x f(a)-a f(x)}{x-a}$ equals :
Correct Option: , 2
Solution:
$f^{\prime}(a)=2, f(a)=4$
$\lim _{x \rightarrow a} \frac{x f(a)-a f(x)}{x-a}$
$\Rightarrow \lim _{x \rightarrow a} \frac{f(a)-a f^{\prime}(x)}{1}$(Lopitals rule)
$=f(a)-a f^{\prime}(a)$
$=4-2 a$