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Q.
The function $f(x)=\frac{\log (1+a x)-\log (1-b x)}{x}$ 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
The function $f ( x )$ is not defined at $x =0$ The function $f(x)$ is to be continuous, then,
$f (0) =\displaystyle\lim _{x \rightarrow 0} f(x)=\displaystyle\lim _{x \rightarrow 0}\left(\frac{\log (1+a x)-\log (1-b x)}{x}\right)$
$=\displaystyle\lim _{x \rightarrow 0}\left(\log \left(\frac{1+a x}{a x}\right) \cdot a-\frac{\log (1-b x)(-b)}{-b x}\right)$
$=\log _{e} e . a-\log _{e} e .(-b)= a + b$