Question

Let $f(x+y)=f\left(x\right)f\left(y\right)andf\left(x\right)=1+xg\left(x\right)G\left(x\right)$, where $\underset{x\to 0}{lim}g\left(x\right)=a$ and $\underset{x\to 0}{lim}G\left(x\right)=b$. Then f'(x) is equal

• A. 1+ab
• B. ab
• C. abf(x)
• D. ab+5

Answer 1

The correct option is C abf(x)

$f(x+y)=f\left(x\right)f\left(y\right)andf\left(x\right)=1+xg\left(x\right).G\left(x\right)$, where
$\underset{x\to 0}{lim}g\left(x\right)=a$ and $\underset{x\to 0}{lim}G\left(x\right)=b$.Then
$f^{\prime }\left(x\right)=\underset{h\to 0}{lim}\frac{f(x+h)-f\left(x\right)}{h}$
$=\underset{h\to 0}{lim}\frac{f\left(x\right)f\left(h\right)-f\left(x\right).1}{h}$
$\therefore f^{\prime }\left(x\right)=f\left(x\right)\underset{h\to 0}{lim}\frac{f\left(h\right)-1}{h}$
$=f\left(x\right)\underset{h\to 0}{lim}g\left(h\right)G\left(h\right)$
$=f\left(x\right)\underset{h\to 0}{lim}g\left(h\right)\underset{h\to 0}{lim}G\left(h\right)$
$=f\left(x\right).a.b.$