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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
On one hand, the limit as n approaches infinity of a sequence {a n} is simply the limit at infinity of a function a(n) —defined on the natural numbers {n}. On the other hand, if X is the domain of a function f ( x ) and if the limit as n approaches infinity of f ( x n ) is L for every arbitrary sequence of points { x n } in X − x 0 which ...
The limit of f as x approaches infinity is L, ... For functions on the real line, one way to define the limit of a function is in terms of the limit of sequences ...
A limit of a sequence of points () in a topological space is a special case of a limit of a function: the domain is in the space {+}, with the induced topology of the affinely extended real number system, the range is , and the function argument tends to +, which in this space is a limit point of .
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 ...
The general form of L'Hôpital's rule covers many cases. Let c and L be extended real numbers: real numbers, positive or negative infinity. Let I be an open interval containing c (for a two-sided limit) or an open interval with endpoint c (for a one-sided limit, or a limit at infinity if c is infinite).
The second limit follows from the squeeze theorem and the fact that 0 ≤ 1 − cos x x ≤ x {\displaystyle 0\leq {\frac {1-\cos x}{x}}\leq x} for x close enough to 0. This can be derived by replacing sin x in the earlier fact by 1 − cos 2 x {\textstyle {\sqrt {1-\cos ^{2}x}}} and squaring the resulting inequality.
The domain of f and g can be any set for which the limit is defined: e.g. real numbers, complex numbers, positive integers. The same notation is also used for other ways of passing to a limit: e.g. x → 0, x ↓ 0, | x | → 0. The way of passing to the limit is often not stated explicitly, if it is clear from the context.