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

• A. $4$
• B. $2$
• C. $1$
• D. $0$

Answer 1

Answer: (A)

Applying L. Hospital’s Rule
$\underset{x\to 2a}{\lim }$ $\frac{f\left(a\right)g^{\prime }\left(a\right)-g\left(a\right)f^{\prime }\left(a\right)}{g^{\prime }\left(a\right)-f^{\prime }\left(a\right)}=4$
$\frac{k\left(g^{\prime }\left(a\right)-f{f}^{\prime }\left(a\right)\right)}{\left(g^{\prime }\left(a\right)-f^{\prime }\left(a\right)\right)}=4$
$k=4.$