Question

If f(a) = 2, f’(a) = 1, g(a) = –1, g’(a) = 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. $\frac{2}{5}$
• D. $1$

Answer 1

The correct option is A $5$

$\underset{x\to a}{lim}\frac{g\left(x\right)f\left(a\right)-g\left(a\right)f\left(x\right)}{x-a}$
$\underset{h\to 0}{lim}\frac{g(a+h)f\left(a\right)-g\left(a\right)f\left(a\right)+g\left(a\right)f\left(a\right)-g\left(a\right)f(a+h)}{h}$
[for x = a + h]
$\underset{h\to 0}{lim}\left[f\left(a\right)\left(\frac{g(a+h)-g\left(a\right)}{h}\right)\right]-g\left(a\right)\left[\frac{f(a+h)-f\left(a\right)}{h}\right]$
$=f\left(a\right)g^{\prime }\left(a\right)–g\left(a\right)f^{\prime }\left(a\right)=2\times 2–(–1)\times 1=5$