List of limits

The following is a compilation of some elementary computations of limits.

By way of notation, $f,\; g$ denote real functions of a real variable, and $a\_n,\; b\_n$ denote sequences of real numbers. For functions, we can have limits either at a real number $a$, in which case it may be either one- or two-sided, or at $pm\; infty$; unless otherwise noted, we have

:$lim\_\{x\; o\; a\}\; f(x)\; =\; L,\; lim\_\{x\; o\; a\}\; g(x)\; =\; M$

as shorthand for both kinds of limits. For sequences, limits are taken only at infinity:

:$lim\_\{n\; o\; infty\}\; a\_n\; =\; A,\; lim\_\{n\; o\; infty\}\; b\_n\; =\; B.$

Properties of limits

Linearity:For any real $s,\; t$, we have::$lim\_\{x\; o\; a\}\; [sf(x)\; +\; tg(x)]\; =\; sL\; +\; tM$::$lim\_\{n\; o\; infty\}\; [s\; a\_n\; +\; t\; b\_n]\; =\; sA\; +\; tB$

Products::$lim\_\{x\; o\; a\}\; f(x)\; g(x)\; =\; LM$::$lim\_\{n\; o\; infty\}\; a\_n\; b\_n\; =\; AB$

Quotients:If $M$ (respectively, $B$) is nonzero, then::$lim\_\{x\; o\; a\}\; frac\{f(x)\}\{g(x)\}\; =\; frac\{L\}\{M\}$::$lim\_\{n\; o\; infty\}\; frac\{a\_n\}\{b\_n\}\; =\; frac\{A\}\{B\}$

:When $N\; =\; 0$ and $M\; eq\; 0$, or if $B\; =\; 0$ and $A\; eq\; 0$, then the limits are $(sgn\{M\})\; infty$ and $(sgn\{A\})infty$, respectively, where $sgn$ is the sign of the number.

Ordering:If $f(x)\; leq\; g(x)$ for all $x$, then $L\; leq\; M.$

:If $a\_n\; leq\; b\_n$ for all $n$, then $A\; leq\; B.$

Local nature:If $f(x)\; =\; g(x)$ for all $x$ sufficiently close to $a$, then $L\; =\; M$.

:If $a\_n\; =\; b\_n$ for all $n$ sufficiently large, then $A\; =\; B$.

Subsequences:If $b\_n$ is a subsequence of $a\_n$, then $A\; =\; B$.

Interlacing:If $A\; =\; B$, the limit of the sequence $a\_1,\; b\_1,\; a\_2,\; b\_2,\; dots$, or in other words of the sequence $c\_n$ with $c\_\{2m\; -\; 1\}\; =\; a\_m,\; c\_\{2m\}\; =\; b\_m$, is::$lim\_\{n\; o\; infty\}\; c\_n\; =\; A\; =\; B$

Supremum and infimum:If $a\_n$ is bounded above, then its limit superior exists and is equal to the supremum of the elements of the sequence::::Furthermore, if $a\_n$ is an increasing sequence then::$limsup\_\{n\; o\; infty\}\; a\_n\; =\; lim\_\{n\; o\; infty\}\; a\_n$

:The analogous statement holds for limits inferior and infima.

Continuity:If $f$ is continuous at $A$, then::$lim\_\{n\; o\; infty\}\; f(a\_n)\; =\; f(A)$

Examples of limits

:$lim\_\{x\; o\; 3\}\; x^2\; =\; 9$:$lim\_\{x\; o\; 0+\}\; x^x\; =\; 1$:$lim\_\{x\; o\; infty\}\; frac\{1\}\{x\}\; =\; 0$:$lim\_\{x\; o\; 0\}\; frac\{sin\{x\{x\}\; =\; 1$:$lim\_\{x\; o\; 0\}\; cos(cx)^\{2/x^2\}\; =\; e^\{-c^2\}$:$lim\_\{n\; o\; infty\}\; 2^\{1/n\}\; =\; lim\_\{n\; o\; infty\}\; sqrt\; [n]\; \{2\}\; =\; 1$:$lim\_\{n\; o\; infty\}\; n^\{1/n\}\; =\; 1$:$lim\_\{n\; o\; infty\}\; frac\{2^n\}\{n!\}\; =\; 0$:$lim\_\{n\; o\; infty\}\; left(1\; +\; frac\{1\}\{n\}\; ight)^\{n\}\; =\; e$

ee also

*Limit of a function
*Limit of a sequence
*Zipper theorem