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 $\underset{x\to a}{\lim }\frac{f\left(a\right)g\left(x\right)-f\left(a\right)-g\left(a\right)f\left(x\right)+f\left(a\right)}{g\left(x\right)-f\left(x\right)}=4$
then the value of k is

• A. 0
• B. 4
• C. 2
• D. 1

Answer 1

Answer: (B)

$\underset{x\to a}{\lim }\frac{f\left(a\right)g^{\prime }\left(x\right)-g\left(a\right)f^{\prime }\left(x\right)}{g^{\prime }\left(x\right)-f^{\prime }\left(x\right)}=4$
(By L’ Hospital rule)
$\underset{x\to a}{\lim }\frac{k{g}^{\prime }\left(x\right)-k{f}^{\prime }\left(x\right)}{g^{\prime }\left(x\right)-f^{\prime }\left(x\right)}=4\therefore k=4$