Question

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

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

Answer 1

Answer: (B)

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