Question

Let $f (a) = g(a) = k$ and their nth derivatives $f^n (a), g^n (a)$ exist and are not equal for some n. Further if $\displaystyle\lim_{x \to a} \frac{f\left(a\right)g\left(x\right) -f\left(a\right) -g\left(a\right)f\left(x\right)+f\left(a\right)}{g\left(x\right)-f\left(x\right)} = 4 $
then the value of k is

• A. 0
• B. 4
• C. 2
• D. 1

Answer 1

Answer: (B)

$\displaystyle\lim_{x \to a} \frac{f\left(a\right)g'\left(x\right) -g\left(a\right)f'\left(x\right)}{g'\left(x\right)-f'\left(x\right)} = 4$
(By L’ Hospital rule)
$\displaystyle \lim_{x \to a} \frac{k g'\left(x\right) -kf'\left(x\right)}{g'\left(x\right)-f'\left(x\right)} =4 \therefore k = 4 $