is an equivalence relation

Note that $\sim$ as defined in the entry Landau notation is an equivalence relation on the set of all functions from $\mathbb{R}^+$ to $\mathbb{R}^+$ . This set of functions will be denoted in this entry as $F$ .

Reflexive: For any $f \in F$ , $\displaystyle \lim_{x \to \infty} \frac{f(x)}{f(x)}=1$ , and $f \sim f$ .

Symmetric: If $f,g \in F$ with $f \sim g$ , then $\displaystyle \lim_{x \to \infty} \frac{f(x)}{g(x)}=1$ . Thus:

Therefore, $g \sim f$ .

Transitive: If $f,g,h \in F$ with $f \sim g$ and $g \sim h$ , then $\displaystyle \lim_{x \to \infty} \frac{f(x)}{g(x)}=1$ and $\displaystyle \lim_{x \to \infty} \frac{g(x)}{h(x)}=1$ . Thus:

Therefore, $f \sim h$ .

is an equivalence relation is owned by Warren Buck.