Question

$\underset{x\to 0}{lim}\frac{a^x-b^x}{x}$=

• A. $\log \left(\frac{a}{b}\right)$
• B. $\log \left(\frac{b}{a}\right)$
• C. $\log \left(ab\right)$
• D. ${\log }_{b}a$

Answer 1

The correct option is A $\log \left(\frac{a}{b}\right)$

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

$=\underset{x\to 0}{lim}\frac{a^x-1-b^x+1}{x}$

$=\underset{x\to 0}{lim}\frac{(a^x-1)-(b^x-1)}{x}$

$=\log a-\log b$ ......... Using formula

$=\log \left(\frac{a}{b}\right)$

Hence, $\underset{x\to 0}{lim}\frac{a^x-b^x}{x}=\log \left(\frac{a}{b}\right)$