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# The function $\Large\frac{log(1+ax)-log(1-bx)}{x}$ is not defined at x = 0. Find the value of f(x) so that f(x) is continuous at x = 0.

Toolbox:
• For $f(x)$ to be continuous at $'0' LHL = RHL = f(0)$
• $\lim\limits_{x \to 0} \; \large\frac{log(1+x)}{x}=1$
LHL = RHL
$= \lim\limits_{x \to 0}\: \large\frac{log(1+ax)-log(1-bx)}{x}$

$= \lim\limits_{x \to 0}\: \large\frac{log(1+ax)}{ax}$x $a-\lim\limits_{x \to 0}\: \large\frac{log(1-bx)}{x(-b)}$ x $(-b)$
$= a+b$
$f(o) = a+b\: and$
$f(x)= \left\{ \begin{array}{1 1} \Large\frac{log(1+ax)-log(1-bx)}{x} & \quad ,\;x\neq0 \\ a+b & \quad ,\;x=0 \end{array} \right.$

edited Mar 26, 2013