Question

Let $f\left(a\right)=g\left(a\right)=k$ and their n${}^{th}$ derivative $f^n\left(a\right)$, $g^n\left(a\right)$ exist and are not equal to 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 $k=$

• A. 2
• B. 1
• C. 0
• D. 4

Answer 1

The correct option is D 4

$\underset{x\to a}{lim}\frac{f\left(a\right)g\left(x\right)-f\left(a\right)-g\left(a\right)f\left(x\right)+g\left(a\right)}{g\left(x\right)-f\left(x\right)}=4$

$\Rightarrow$ $\underset{x\to a}{lim}\frac{f\left(a\right)[g\left(x\right)-f\left(x\right)]}{g\left(x\right)-f\left(x\right)}=4$

$\Rightarrow$ $\underset{x\to a}{lim}f\left(a\right)=4$

$\Rightarrow k=4$