Question

The function $f\left(x\right)=\frac{ln(1+ax)-ln(1-bx)}{x}$ not defined at $x=0$. The value which should be assigned to $f$ at $x=0$ so that it is continuous as $x=0$, is

• A. $a-b$
• B. $a+b$
• C. $lna+lnb$
• D. None of these

Answer 1

The correct option is B $a+b$

For $f\left(x\right)$ to be continuous, we must have
$f\left(0\right)=\underset{x\to 0}{lim}f\left(x\right)$
$\therefore \underset{x\to 0}{lim}\frac{\log (1+ax)-\log (1-bx)}{x}$
$=\underset{x\to 0}{lim}\frac{a\log (1+ax)}{ax}+\frac{b\log (1-bx)}{-bx}$
$=a\cdot 1+b\cdot 1\left[using\underset{x\to 0}{lim}\frac{\log (1+x)}{x}-1\right]$
$=a+b$
$\therefore f\left(0\right)=a+b$.