Asymptotically equivalent
High school
Functions $f$ and $g$ are asymptotically equivalent as $x \to \infty$ if the ratio $f(x) / g(x)$ approaches $1$, so that $f(x)$ and $g(x)$ are nearly equal in relative terms for large $x$.
Undergraduate
Functions $f$ and $g$ are asymptotically equivalent as $x \to \infty$, written $f(x) \sim g(x)$, if \(\lim_{x \to \infty} f(x) / g(x) = 1.\)
Examples
Non-examples
Things that may be mistaken for examples.
- $f(x) = x^2$, $g(x) = x$ Undergraduate
Properties
- If $f \sim g$, then $f = \Theta(g)$, but not conversely.
- $f \sim g$ if and only if $f - g = o(g)$.