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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
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B
a+b
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C
ln a+ln b
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D
None of these
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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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