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This is a list of limits for common functions such as elementary functions. In this article, the terms a , b and c are constants with respect to x . Limits for general functions
In mathematics, the nth-term test for divergence [1] is a simple test for the divergence of an infinite series: If lim n → ∞ a n ≠ 0 {\displaystyle \lim _{n\to \infty }a_{n}\neq 0} or if the limit does not exist, then ∑ n = 1 ∞ a n {\displaystyle \sum _{n=1}^{\infty }a_{n}} diverges.
Limits can be difficult to compute. There exist limit expressions whose modulus of convergence is undecidable. In recursion theory, the limit lemma proves that it is possible to encode undecidable problems using limits. [15] There are several theorems or tests that indicate whether the limit exists. These are known as convergence tests.
respectively. If these limits exist at p and are equal there, then this can be referred to as the limit of f(x) at p. [9] If the one-sided limits exist at p, but are unequal, then there is no limit at p (i.e., the limit at p does not exist). If either one-sided limit does not exist at p, then the limit at p also does not exist.
exists there are three possibilities: if L > 1 the series converges (this includes the case L = ∞) if L < 1 the series diverges; and if L = 1 the test is inconclusive. An alternative formulation of this test is as follows. Let { a n} be a series of real numbers. Then if b > 1 and K (a natural number) exist such that
In multivariable calculus, an iterated limit is a limit of a sequence or a limit of a function in the form , = (,), (,) = ((,)),or other similar forms. An iterated limit is only defined for an expression whose value depends on at least two variables. To evaluate such a limit, one takes the limiting process as one of the two variables approaches some number, getting an expression whose value ...
In other words, since the two one-sided limits exist and are equal, the limit of () as approaches exists and is equal to this same value. If the actual value of f ( x 0 ) {\displaystyle f\left(x_{0}\right)} is not equal to L , {\displaystyle L,} then x 0 {\displaystyle x_{0}} is called a removable discontinuity .
For any continuous function, if exists, then () exists too. In fact, any real-valued function f {\textstyle f} is continuous if and only if it preserves the limits of sequences (though this is not necessarily true when using more general notions of continuity).
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