L'Hôpital's Rule/Examples/(Root (1 + x) - 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
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): l'Hôpital's rule
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): l'Hôpital's rule