Question

If $f\left(a\right)=3,f^{\prime }\left(a\right)=-2,g\left(a\right)=-1,g^{\prime }\left(a\right)=4$, then $\underset{x\to a}{lim}\frac{g\left(x\right)f\left(a\right)-g\left(a\right)f\left(x\right)}{x-a}$ equals

• A. $-5$
• B. $10$
• C. $-10$
• D. $5$

Answer 1

The correct option is B $10$

Using L-hospital's rule,
$\underset{x\to a}{lim}\frac{g\left(x\right)f\left(a\right)-g\left(a\right)f\left(x\right)}{x-a}=\underset{x\to a}{lim}\frac{g^{\prime }\left(x\right)f\left(a\right)-g\left(a\right)f^{\prime }\left(x\right)}{1}$
$=g^{\prime }\left(a\right)f\left(a\right)-g\left(a\right)f^{\prime }\left(a\right)=4\times 3-(-1)\times (-2)=10$
Hence, option 'B' is correct.