In mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value. The concept of limit allows mathematicians to define a new point from a Cauchy sequence of previously defined points within a complete metric space. Limits are essential to calculus (and mathematical analysis in general) and are used to define continuity, derivatives, and integrals.
The concept of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closely related to limit and direct limit in category theory.
In formulas, limit is usually abbreviated as lim as in lim(an) = a or represented by the right arrow (→) as in an → a.
The concept of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closely related to limit and direct limit in category theory.
In formulas, limit is usually abbreviated as lim as in lim(an) = a or represented by the right arrow (→) as in an → a.
Although implicit in the development of calculus of the 17th and 18th centuries, the modern notion of the limit of a function goes back to Bolzano who, in 1817, introduced the basics of the epsilon-delta technique to define continuous functions. However, his work was not known during his lifetime (Felscher 2000). Cauchy discussed limits in his Cours d'analyse (1821) and gave essentially the modern definition, but this is not often recognized because he only gave a verbal definition (Grabiner 1983). Weierstrass first introduced the epsilon-delta definition of limit in the form it is usually written today. He also introduced the notations lim and limx→x0 (Burton 1997).
The modern notation of placing the arrow below the limit symbol is due to Hardy in his book A Course of Pure Mathematics in 1908 (Miller 2004).
The modern notation of placing the arrow below the limit symbol is due to Hardy in his book A Course of Pure Mathematics in 1908 (Miller 2004).
Imagine a person walking over a landscape represented by the graph of y = f(x). His horizontal position is measured by the value of x, much like the position given by a map of the land or by a global positioning system. His altitude is given by the coordinate y. He is walking towards the horizontal position given by x = p. As he does so, he notices that his altitude approaches L. If later asked to guess the altitude over x = p, he would then answer L, even if he had never actually reached that position.
What, then, does it mean to say that his altitude approaches L? It means that his altitude gets nearer and nearer to L except for a possible small error in accuracy. For example, suppose a particular accuracy goal is set for our traveler: he must get within ten meters of L. He reports back that indeed he can get within ten meters of L, since he notes that when he is within fifty horizontal meters of p, his altitude is always ten meters or less from L.
The accuracy goal is then changed: can he get within one meter? Yes. If he is within seven horizontal meters of p, then his altitude remains within one meter of the target L. In summary, to say that the traveler's altitude approaches L as his horizontal position approaches p means that for every target accuracy goal, there is some neighborhood of p whose altitude remains within that accuracy goal.
The initial informal statement can now be explicated:
The limit of a function f(x) as x approaches p is a number L with the following property: given any target distance from L, there is a distance from p within which the values of f(x) remain within the target distance.
This explicit statement is quite close to the formal definition of the limit of a function with values in a topological space.
What, then, does it mean to say that his altitude approaches L? It means that his altitude gets nearer and nearer to L except for a possible small error in accuracy. For example, suppose a particular accuracy goal is set for our traveler: he must get within ten meters of L. He reports back that indeed he can get within ten meters of L, since he notes that when he is within fifty horizontal meters of p, his altitude is always ten meters or less from L.
The accuracy goal is then changed: can he get within one meter? Yes. If he is within seven horizontal meters of p, then his altitude remains within one meter of the target L. In summary, to say that the traveler's altitude approaches L as his horizontal position approaches p means that for every target accuracy goal, there is some neighborhood of p whose altitude remains within that accuracy goal.
The initial informal statement can now be explicated:
The limit of a function f(x) as x approaches p is a number L with the following property: given any target distance from L, there is a distance from p within which the values of f(x) remain within the target distance.
This explicit statement is quite close to the formal definition of the limit of a function with values in a topological space.
Source: http://www.wikipedia.org
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