Question

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

• A.
• B.
• C.
• D.

Answer 1

The correct option is B

We have, $\underset{x\to 0}{lim}\left(\frac{a^x-b^x}{x}\right)$

At x=0 the value of the given expression takes $\frac{0}{0}$ form

Making the given expression of standard limits

by adding and subtracting 1

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

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

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

we know,

$=\underset{x\to 0}{lim}\frac{a^x-1}{x}=lo{g}_{e}a$

So,$=lo{g}_{e}a-lo{g}_{e}b$

$=lo{g}_e\left(\frac{a}{b}\right)$

Answer option B is correct