Let ƒ(x) be a differentiable function

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 :

  1. $2 a+4$

  2. $4-2 a$

  3. $2 a-4$

  4. $a+4$


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)

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

$=4-2 a$

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