- 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 thesupremum of the elements of the sequence:::limsup_{n o infty} a_n = sup_n{a_n} < infty:Furthermore, if a_n is anincreasing 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
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