Question

$\underset{x\to 0}{lim}\frac{{\left(ab\right)}^x-a^x-b^x+1}{x^2}$

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

Answer 1

The correct option is A $\log a\cdot \log b$

Given: ${lim}_{x\to 0}\frac{({ab)}^x-a^x-b^x+1}{x^2}$

To find: Value of limit

Sol: $l={lim}_{x\to 0}\frac{a^x.b^x-a^x-b^x+1}{x^2}$

$={lim}_{x\to 0}\frac{{(a}^x-1\left)\right(b^x-1)}{x^2}$

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

$=\log \left(a\right).log\left(b\right)$ is the correct answer