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Limits involving infinity are connected with the concept of asymptotes. These notions of a limit attempt to provide a metric space interpretation to limits at infinity. In fact, they are consistent with the topological space definition of limit if
The graph of this function has a horizontal asymptote at = ... enables a definition of "limits at infinity" which is very similar to the usual defininion of limits, ...
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 ...
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.
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.
has a limit of +∞ as x → 0 +, ƒ(x) has the vertical asymptote x = 0, even though ƒ(0) = 5. The graph of this function does intersect the vertical asymptote once, at (0, 5). It is impossible for the graph of a function to intersect a vertical asymptote (or a vertical line in general) in more than one point.
A limit taking one of these indeterminate forms might tend to zero, might tend to any finite value, might tend to infinity, or might diverge, depending on the specific functions involved. A limit which unambiguously tends to infinity, for instance lim x → 0 1 / x 2 = ∞ , {\textstyle \lim _{x\to 0}1/x^{2}=\infty ,} is not considered ...
In mathematical analysis, limit superior and limit inferior are important tools for studying sequences of real numbers.Since the supremum and infimum of an unbounded set of real numbers may not exist (the reals are not a complete lattice), it is convenient to consider sequences in the affinely extended real number system: we add the positive and negative infinities to the real line to give the ...