Question

Evaluate :$\underset{x\to 0}{lim}\frac{2^x-1}{{(1+x)}^{1/2}-1}$

Answer 1

We have,

$\underset{x\to 0}{lim}\frac{2^x-1}{{(1+x)}^{1/2}-1}$

This is $\frac{0}{0}$ form.

So, apply L-Hospital rule,

$\underset{x\to 0}{lim}\frac{2^x\ln 2-0}{\frac{1}{2\sqrt{1+x}}-0}$

$\underset{x\to 0}{lim}{2}^x\ln 2\left(2\sqrt{1+x}\right)$

$=2^0\ln 2\left(2\sqrt{1+0}\right)$

$=1\times \ln 2\left(2\sqrt{1}\right)$

$=2\ln 2$

Hence, this is the answer.