Question

$ \lim_{x \to {0}} \frac {a^x-b^x} { x } $is

• A. log ab
• B. $log \frac {a}{b} $
• C. $\frac {log a} {log b}$
• D. $\frac {log b} {log a}$

Answer 1

Answer: (B)

$\lim_{x\to0} \frac{a^{x} - b^{x}}{x} =\lim_{x\to0} \frac{\left(a^{x} -1 \right) -\left(b^{x} -1\right)}{x} $
= $\lim_{x\to0} \frac{a^{x} -1}{x} - \lim _{x\to 0} \frac{b^{x } -1}{x}$
= $\log \, a - \log\, b = \log\left(\frac{a}{b}\right)$