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Question
Let \[f\left( x \right) = \frac{\log\left( 1 + \frac{x}{a} \right) - \log\left( 1 - \frac{x}{b} \right)}{x}\] x ≠ 0. Find the value of f at x = 0 so that f becomes continuous at x = 0.
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Solution
Given:
If f(x) is continuous at x = 0, then
\[\Rightarrow \frac{1}{a} \times 1 - \left( - \frac{1}{b} \right) \times 1 = f\left( 0 \right) \left[ \text{ Using } : \lim_{x \to 0} \frac{\log\left( 1 + x \right)}{x} = 1 \right]\]
\[ \Rightarrow \frac{1}{a} + \frac{1}{b} = f\left( 0 \right)\]
\[ \Rightarrow \frac{a + b}{ab} = f\left( 0 \right)\]
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