$\sim$ 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 ${ℝ}^{+}$ to ${ℝ}^{+}$. This set of functions will be denoted in this entry as $F$.

Reflexive (http://planetmath.org/Reflexive): For any $f\in F$, $\underset{x\to \mathrm{\infty }}{lim}{\displaystyle \frac{f(x)}{f(x)}}=1$, and $f\sim f$.

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

Therefore, $g\sim f$.

Transitive (http://planetmath.org/Transitive3): If $f,g,h\in F$ with $f\sim g$ and $g\sim h$, then $\underset{x\to \mathrm{\infty }}{lim}{\displaystyle \frac{f(x)}{g(x)}}=1$ and $\underset{x\to \mathrm{\infty }}{lim}{\displaystyle \frac{g(x)}{h(x)}}=1$. Thus:

Therefore, $f\sim h$.