Asymptotic Equivalence¶

Definition¶

Let $f$ and $g$ be defined near $a\text{,}$ with $g(x)\neq0$ sufficiently near $a\text{.}$ We say that $f$ and $g$ are asymptotically equivalent as $x\to a\text{,}$ and write $f(x)\sim g(x)\text{,}$ if

Interpretation¶

Asymptotic equivalence describes when two functions behave almost identically near a given point or at infinity, up to a negligible relative error. It compares sizes by their ratio, not by their difference: $f\sim g$ means that $f(x)=g(x)\bigl(1+r(x)\bigr)$ with $r(x)\to0$$0$.

Near $0$ the graphs of $\sin x$ and $x$ almost coincide, and the ratio $\sin x/x$ tends to $1$, so $\sin x\sim x$ as $x\to0$$0$. The ratio is not defined at $x=0$ itself; only the behaviour close to $0$ matters.

The ratio is evaluated by a limit at a point for finite $a$, or a limit at infinity for $a=+\infty$ or $-\infty$. The extended real numbers allow these infinite approach points; the ratio itself still tends to the finite number 1.

Many equivalences come from derivatives: if $f(a)=0$ and $f'(a)\neq0$$0$, then $f(x)\sim f'(a)(x-a)$ as $x\to a$$a$, because the ratio is the difference quotient of $f$ divided by $f'(a)$. This gives $\mathrm{e}^x-1\sim x$ and $\ln(1+x)\sim x$ as $x\to0$$0$.

Two slips are common. First, $f\sim g$ does not mean that $f-g\to0$$0$: $x^2+x\sim x^2$ as $x\to\infty$ although the difference tends to $\infty$, while as $x\to0$ the difference $2x-x$ tends to $0$ but $2x\not\sim x$$x$. Second, equivalent functions may be substituted in products and quotients, but not in sums: replacing $\sin x$ by $x$ in $\sin x-x$ gives $0$, whereas in fact $\sin x-x\sim -x^3/6$ as $x\to0$$0$. Math III refines this comparison of sizes with the big-$O$ and little-$o$ notations.

Key Results¶

• Asymptotic equivalence is symmetric (i.e., $f\sim g\iff g\sim f$$f$) and transitive (i.e., $f\sim g, \,g\sim h\implies f \sim h$$h$).

• Operations that preserve equivalence. Assume $f_1\sim g_1$ and $f_2\sim g_2$ as $x\to a\text{.}$ Whenever the expressions are defined,

  • $f_1(x)\cdot f_2(x)\sim g_1(x)\cdot g_2(x)$ as $x\to a\text{;}$
  • $f_1(x)/f_2(x)\sim g_1(x)/g_2(x)$ as $x\to a\text{;}$
  • $|f_1(x)|^r\sim |g_1(x)|^r$ for every $r\in\mathbb{R}$ as $x\to a\text{;}$
  • if $f_1,g_1>0$ and both tend to $0\text{,}$ or both tend to $+\infty\text{,}$ then $\ln f_1(x)\sim\ln g_1(x)\text{.}$

• Addition and subtraction require extra care, because leading terms may cancel.

• Equivalent functions may therefore be substituted for one another in limits involving products or quotients.

Examples¶

• $x\sim\sin x\sim(\mathrm{e}^x-1)\sim\ln(x+1)$ as $x\to0\text{.}$
• $1-\cos x\sim x^2/2$ as $x\to 0\text{.}$

Prerequisites¶

• Limit at a Point
• Limit at Infinity
• Extended Real Number

Related Concepts¶

• Big-$O$ and Little-$o$ Notations
• Division and Quotient
• Indeterminate Form
• Limit Laws for Functions
• Squeeze Theorem for Functions

Sources¶

• Math I: Calculus · Lecture 8

Aliases: asymptotically equivalent functions, asymptotically equivalent Keywords: puppy rule Symbols: element of, infinity, real numbers

Last updated: 20 October 2026 03:02