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QUESTION

Using l'Hopital's rule, or otherwise, evaluate
$\ds\lim_{x\to0}\left(\frac{(1+x)^\frac{1}{2}-1}{x}\right)$.

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ANSWER

$\ds\lim_{x\to0}\left(\frac{(1+x)^\frac{1}{2}-1}{x}\right)=\frac{1-1}{0}=\frac{0}{0}?\\
=\lim_{x\to0}\left(\frac{\frac{1}{2}(1+x)^{-\frac{1}{2}}}{1}\right)=
\frac{\frac{1}{2}(1)^{-\frac{1}{2}}}{1}=\frac{1}{2}$





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