Question

Use the formula $li{m}_{x\to 0}\frac{a^x-1}{x}=lo{g}_{e}a,$ to find
$li{m}_{x\to 0}\frac{2^x-1}{(1+x{)}^{1/2}-1}$.

Answer 1

$li{m}_{x\to 0}\frac{2^x-1}{\sqrt{1+x}-1}\times \frac{\sqrt{1+x}+1}{\sqrt{1+x}+1}=li{m}_{x\to 0}\frac{(2^x-1)(\sqrt{1+x}+1)}{x}$
= $lo{g}_e$ (2) . (2)
= 2 $lo{g}_{e}2=lo{g}_{e}4$.