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 $\underset{x\to a}{lim}\frac{g\left(x\right)f\left(a\right)-g\left(a\right)f\left(x\right)}{x-a}$ is?

• A. $-5$
• B. $\frac{1}{5}$
• C. $5$
• D. $-\frac{1}{5}$

Answer 1

The correct option is C $5$

$\underset{x\to a}{lim}\frac{g\left(x\right)f\left(a\right)-g\left(a\right)f\left(x\right)}{x-a}$
Applying L-Hospital's rule,
$={lim}_{x\to a}\frac{f\left(a\right)g^{{}^{\prime }}\left(x\right)-g\left(a\right)f^{{}^{\prime }}\left(x\right)}{1}$
$=4-(-1)=5$
Hence, option 'C' is correct.