Question

# The function $$f(x) = \dfrac {ln (1 + ax) - ln(1 - bx)}{x}$$ 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
ab
B
a+b
C
ln a+ln b
D
None of these

Solution

## The correct option is B $$a + b$$For $$f(x)$$ to be continuous, we must have$$f(0) = \displaystyle \lim_{x\rightarrow 0} f(x)$$$$\therefore \displaystyle \lim_{x\rightarrow 0} \dfrac {\log (1 + ax) - \log (1 - bx)}{x}$$$$= \displaystyle \lim_{x\rightarrow 0} \dfrac {a\log (1 + ax)}{ax} + \dfrac {b\log (1 - bx)}{-bx}$$$$= a\cdot 1 + b\cdot 1 \left [using \displaystyle \lim_{x\rightarrow 0} \dfrac {\log (1 + x)}{x} - 1\right ]$$$$= a + b$$$$\therefore f(0) = a + b$$.Mathematics

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