List of limits

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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 an,bn 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.

[edit] 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 an = bn for all n sufficiently large, then A = B.

Subsequences

If bn is a subsequence of an, 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 cn with c2m − 1 = am,c2m = bm, is
\lim_{n \to \infty} c_n = A = B

Supremum and infimum

If an 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 an 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)

[edit] 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_{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{1}{n}\right)^{n} = e
\lim_{n \to 0} cos(nx)^{2/n^2} = e^{-x^2}

[edit] See also