Definition · applies to functions · also called $f \sim g$

High school

Functions $f$ and $g$ are asymptotically equivalent as $x \to \infty$ if their ratio $f(x) / g(x)$ approaches $1$, so that they 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 \(\lim_{x \to \infty} f(x) / g(x) = 1.\)

Properties

  • If $f \sim g$, then $f = \Theta(g)$, but not conversely.
  • $f \sim g$ if and only if $f - g = o(g)$.

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