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)+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
$\displaystyle \lim_{x \to 2a}$ $\frac{f \left(a\right)g'\left(a\right)-g\left(a\right)f '\left(a\right)}{g'\left(a\right)-f '\left(a\right)}=4$
$\frac{k\left(g'\left(a\right)-ff '\left(a\right)\right)}{\left(g'\left(a\right)-f '\left(a\right)\right)}=4$
$k=4.$