A) \[\log \left( \frac{a}{b} \right)\]
B) \[\log \left( \frac{b}{a} \right)\]
C) \[\log (a\,b)\]
D) \[\log \,(a+\,b)\]
Correct Answer: A
Solution :
\[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{a}^{x}}-{{b}^{x}}}{{{e}^{x}}-1}=\underset{x\to 0}{\mathop{\lim }}\,\frac{{{a}^{x}}-{{b}^{x}}}{x}.\frac{x}{{{e}^{x}}-1}\] \[=\underset{x\to 0}{\mathop{\lim }}\,\left[ \frac{{{a}^{x}}-1}{x}-\frac{{{b}^{x}}-1}{x} \right]\frac{x}{{{e}^{x}}-1}\] \[=({{\log }_{e}}a-{{\log }_{e}}b).\frac{1}{{{\log }_{e}}e}\]\[={{\log }_{e}}\left( \frac{a}{b} \right)\] Trick : Apply L-Hospital?s rule.
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