Infinitesimals and equivalences

Level UniversityDifficulty ★★★★★Concept⌖ Open in the map

What is it?

A quantity that tends to 0. Two infinitesimals are equivalent ( $f \sim g$ ) if $f/g \to 1$ : near 0, $\sin x \sim x$ , $e^x - 1 \sim x$ , $\ln(1 + x) \sim x$ . Replacing one by the other simplifies limits — and, in floating point, avoids catastrophic cancellation.

Formulas

f \sim g \ (x \to a) \iff \lim_{x\to a}\frac{f(x)}{g(x)} = 1

\sin x \sim x, \quad 1 - \cos x \sim \tfrac{x^2}{2}, \quad e^x - 1 \sim x, \quad \ln(1+x) \sim x

Where it shows up in computing

Floating point (IEEE 754)★★★★★frequentScientific computing and algorithms
expm1(x) and log1p(x) exist because computing $e^x - 1$ or $\ln(1 + x)$ directly loses all precision for tiny $x$ .

Where is it used?

Computing topics reachable from here, through the chain of ideas that leads to them:

λ Scientific computing and algorithms

Floating point (IEEE 754)★★★★★
L'Hôpital's rule→Algorithm analysis and complexity★★★★★
Floating point (IEEE 754)→Scientific computing★★★★★

What depends on it

L'Hôpital's rule

This page has the essentials. A fuller treatment (intuition, formal definition, worked example) is on the way.