Question

Let $f\left(x\right)$ be a differentiable function at $x=a$ with $f\text{'}\left(a\right)=2$ and $f\left(a\right)=4$. Then $\underset{x\to a}{\lim }\left[\frac{xf\left(a\right)-af\left(x\right)}{x-a}\right]$ equals

• A. $2a+4$
• B. $2a-4$
• C. $4-2a$
• D. $a+4$

Answer 1

The correct option is C

$4-2a$

Explanation of correct answer :

Finding value of $\underset{x\to a}{\lim }\left[\frac{xf\left(a\right)-af\left(x\right)}{x-a}\right]$:

Given, $f\text{'}\left(a\right)=2$ and $f\left(a\right)=4$.

Using, L-Hopital rule :

$$\begin{gathered} =\underset{x\to a}{\lim }\left[\frac{f\left(a\right)+xf\text{'}\left(a\right)-af\text{'}\left(x\right)}{1}\right]\left(\text{differentiating both numerator and denominator}\right) \\ =f\left(a\right)-af\text{'}\left(a\right)\left(\text{given}\right) \\ =4-2a \end{gathered}$$
Thus, the value of $\underset{x\to a}{\lim }\left[\frac{xf\left(a\right)-af\left(x\right)}{x-a}\right]$ is $4-2a$.

Hence, Option (C) is correct.