L'Hôpital's Rule/Examples/(Root of (1 plus x) minus 1) over x

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Example of Use of L'Hôpital's Rule

$\ds \lim_{x \mathop \to 0} \dfrac {\sqrt {1 + x} - 1} x = \dfrac 1 2$


Proof

Let $f: \R \to \R$ be defined as:

$\forall x \in \R: \map f x = \sqrt {1 + x} - 1$

Let $g: \R \to \R$ be defined as:

$\forall x \in \R: \map g x = x$


We have that:

\(\ds \map {f'} x\) \(=\) \(\ds \dfrac \d {\d x} \sqrt {1 + x} - 1\)
\(\ds \) \(=\) \(\ds \frac 1 2 \paren {\sqrt {1 + x} }^{-1/2}\) Power Rule for Derivatives
\(\ds \map {g'} x\) \(=\) \(\ds \dfrac \d {\d x} x\)
\(\ds \) \(=\) \(\ds 1\) Power Rule for Derivatives

Then:

\(\ds \lim_{x \mathop \to 0} \dfrac {\map f x} {\map g x}\) \(=\) \(\ds \lim_{x \mathop \to 0} \dfrac {\map {f'} x} {\map {g'} x}\)
\(\ds \) \(=\) \(\ds \lim_{x \mathop \to 0} \dfrac {\frac 1 2 \paren {\sqrt {1 + x} }^{-1/2} } 1\)
\(\ds \) \(=\) \(\ds \lim_{x \mathop \to 0} \frac 1 2 \times 1\)
\(\ds \) \(=\) \(\ds \frac 1 2\)

$\blacksquare$


Sources