Question

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

• A. $a-b$
• B. $a+b$
• C. $\ln a+\ln b$
• D. None of these

Answer 1

Answer: (B)

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