Question

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

• A. $5$
• B. $-5$
• C. $3$
• D. $-3$

Answer 1

The correct option is A $5$

$\underset{x\to a}{lim}\frac{g\left(x\right)f\left(a\right)-g\left(a\right)f\left(x\right)}{x-a},\frac{0}{0}$ form

Use L-hospital rule

$=\underset{x\to a}{lim}\frac{g^{\prime }\left(x\right)f\left(a\right)-g\left(a\right)f^{\prime }\left(x\right)}{1-0}$

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

$=2\cdot 2-(-1)\cdot 1=4+1=5$