List of limits.html

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It has been suggested that this article or section be merged with table of limits. (Discuss)

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

$\lim_{x \to a} f(x) = L, \lim_{x \to a} g(x) = M$

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

$\lim_{n \to \infty} a_n = A, \lim_{n \to \infty} b_n = B.$

Properties of limits

Linearity

For any real

s, t

, we have

$\lim_{x \to a} [sf(x) + tg(x)] = sL + tM$

$\lim_{n \to \infty} [s a_n + t b_n] = sA + tB$

Products

$\lim_{x \to a} f(x) g(x) = LM$

$\lim_{n \to \infty} a_n b_n = AB$

Quotients

If

M

(respectively,

B

) is nonzero, then

$\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{L}{M}$

$\lim_{n \to \infty} \frac{a_n}{b_n} = \frac{A}{B}$

When

N = 0

and $M \neq 0$ , or if

B = 0

and $A \neq 0$ , 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 2 m - 1 = a m, c 2 m = b m

, is

$\lim_{n \to \infty} c_n = A = B$

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:

$\limsup_{n \to \infty} a_n = \sup_n\{a_n\} < \infty$

Furthermore, if

a n

is an increasing sequence then

$\limsup_{n \to \infty} a_n = \lim_{n \to \infty} a_n$

The analogous statement holds for limits inferior and infima.

Continuity

If

f

is continuous at

A

, then

$\lim_{n \to \infty} f(a_n) = f(A)$

Examples of limits

$\lim_{x \to 3} x^2 = 9$

$\lim_{x \to 0+} x^x = 1$

$\lim_{x \to \infty} \frac{1}{x} = 0$

$\lim_{x \to 0} \frac{\sin{x}}{x} = 1$

$\lim_{x \to 0} \cos(cx)^{2/x^2} = e^{-c^2}$

$\lim_{n \to \infty} 2^{1/n} = \lim_{n \to \infty} \sqrt[n]{2} = 1$

$\lim_{n \to \infty} n^{1/n} = 1$

$\lim_{n \to \infty} \frac{2^n}{n!} = 0$

$\lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^{n} = e^{x}$

See also

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