Question

If $f\left(a\right)=2,f^{\prime }\left(a\right)=1,g\left(a\right)=-1$ and $g^{\prime }\left(a\right)=2$. Find the value of $\underset{x\to a}{lim}\frac{g\left(x\right)f\left(a\right)-g\left(a\right)f\left(x\right)}{(x-a)}$

Answer 1

$\underset{x\to a}{lim}\frac{g^{\prime }\left(x\right)f\left(a\right)-g\left(a\right)f^{\prime }\left(x\right)}{1}$ [using L'Hospital rule]

$=g^{\prime }\left(a\right)f\left(a\right)-g\left(a\right)f^{\prime }\left(a\right)$

$=\left(2\right)\left(2\right)-(-1)\left(1\right)=4+1=5$