Question

If $f\left(x\right),g\left(x\right)$ be differentiable function and $f\left(1\right)=g\left(1\right)=2$ then $\underset{x\to 1}{lim}\frac{f\left(1\right)g\left(x\right)-f\left(x\right)g\left(1\right)-f\left(1\right)+g\left(1\right)}{g\left(x\right)-f\left(x\right)}$ is equal to

• A. $0$
• B. $1$
• C. $2$
• D. none of these

Answer 1

The correct option is C $2$

By L'Hospital's Rule,
$=\underset{x\to 1}{lim}\frac{f\left(1\right)g^{\prime }\left(x\right)-g\left(1\right)f^{\prime }\left(x\right)}{g^{\prime }\left(x\right)-f^{\prime }\left(x\right)}$
$=\underset{x\to 1}{lim}\frac{2\left(g^{\prime }\right(x)-f^{\prime }(x\left)\right)}{g^{\prime }\left(x\right)-f^{\prime }\left(x\right)}=2$