Question

Find $\underset{x\to 0}{lim}\left(\frac{3^x-1}{\sqrt{1+x}-1}\right)$

Answer 1

Consider the following question.

${lim}_{x\to 0}\left(\frac{3^x-1}{\sqrt{1+x}-1}\right)$

Using L-Hospital rules,

$\underset{x\to 0}{=lim}\left(\frac{3^x\log 3}{\frac{1}{2\sqrt{x+1}}}\right)$

$={lim}_{x\to }(3^x\log 3\ast 2)\sqrt{x+1}$

$={lim}_{x\to }(3^0\log 3\ast 2)\left(\sqrt{0+1}\right)$

$=2\ln 3$

Hence, this is the required answer.