Question

Evaluate: $\underset{x\to 0}{lim}\frac{6^x-1}{\sqrt{3-x}-\sqrt{3}}$.

Answer 1

$\underset{x\to 0}{lim}\frac{6^x-1}{\sqrt{3-x}-\sqrt{3}}\times \frac{\sqrt{3-x}+\sqrt{3}}{\sqrt{3-x}+\sqrt{3}}$
$\underset{x\to 0}{lim}\frac{(6^x-1)}{(3-x)-\left(3\right)}\cdot (\sqrt{3-x}+\sqrt{3})$
$\underset{x\to 0}{lim}\frac{6^x-1}{3-x-3}\cdot (\sqrt{3-x}+\sqrt{3})$
$\underset{x\to 0}{lim}-\left(\frac{6^x-1}{x}\right)\underset{x\to 0}{lim}(\sqrt{3-x}+\sqrt{3})$
$=-{\log }_{e}6\cdot (\sqrt{3}+\sqrt{3})$
$=-{\log }_{e}6\left(2\sqrt{3}\right)$
$=-2\sqrt{3}{\log }_{e}6$.