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if she is continued in infinity but which she can approach nearer than a given segment 4 in the scholium to principia in 1687 isaac newton had a clear definition of a limit stating that those ultimate ratios are not actually ratios of ultimate quantities but limits which they can approach so closely that their difference is less than any given quantity 5 bruce pourciau further argues that in addition to newton actually having a more sophisticated understanding of limits than he is generally credited with he also provided the first epsilon argument 6 the modern definition of a limit goes back to bernard bolzano who in 1817 developed the basics of the epsilon delta technique to define continuous functions however his work remained unknown to other mathematicians until thirty years after his death 7 augustin louis cauchy in 1821 8 followed by karl weierstrass formalized the definition of the limit of a function which became known as the ε δ definition of limit the modern notation of placing the arrow below the limit symbol was invented by john gaston leathem in 1905 and popularized by g h hardy s 1908 textbook a course of pure mathematics 9 types of limits edit in sequences edit main article limit of a sequence real numbers edit a sequence of real numbers a 0 a 1 displaystyle a_ 0 a_ 1 dots is said to converge to a limit l displaystyle l if the terms of the sequence give an arbitrarily good approximation of the number l displaystyle l after discarding finitely many initial terms an example is the sequence 0 0 3 0 33 0 333 0 3333 displaystyle 0 0 3 0 33 0 333 0 3333 and so on each term in the sequence is an approximation to the rational number 1 3 0 333 displaystyle 1 3 0 333 ldots if an approximation is desired that lies within a given error ϵ displaystyle epsilon then all of the terms of the sequence except finitely many lie within the target error for example to get within ϵ 0 001 displaystyle epsilon 0 001 of 1 3 displaystyle 1 3 the approximations 0 displaystyle 0 0 3 displaystyle 0 3 and 0 33 displaystyle 0 33 are outside the error window but after discarding those three approximations all of the remaining approximations are within the error range ϵ 0 001 displaystyle epsilon 0 001 the same approximation property holds for every positive error ϵ displaystyle epsilon more precisely a sequence a n n n displaystyle a_ n _ n in mathbb n of real numbers is said to be convergent if there exists an l r displaystyle l in mathbb r such that for each positive real number ϵ displaystyle epsilon there exists a positive integer n displaystyle n depending on ϵ displaystyle epsilon such that every term a n displaystyle a_ n with n n displaystyle n n is within distance ϵ displaystyle epsilon of l displaystyle l the number l displaystyle l is unique when it exists it is called the limit of the sequence a n n n displaystyle a_ n _ n in mathbb n and this is written lim n a n l displaystyle lim _ n to infty a_ n l the definition of the limit of a sequence can thus be summarized lim n a n l displaystyle lim _ n to infty a_ n l means that for every ϵ 0 displaystyle epsilon 0 there exists n n displaystyle n in mathbb n such that a n l ϵ displaystyle a_ n l epsilon for all n n displaystyle n n if such an l displaystyle l does not exist a n n n displaystyle a_ n _ n in mathbb n is said to be divergent a sequence is said to tend to infinity if all but finitely many terms exceed each given number an example is the sequence 1 2 3 displaystyle 1 2 3 ldots of positive integers this tends to infinity because for any given bound m displaystyle m every element of the sequence is greater than m displaystyle m after discarding the finite number that are not formally a n n n displaystyle a_ n _ n in mathbb n tends to infinity if given any real m displaystyle m there exists a positive integer n displaystyle n such that a n m displaystyle a_ n m for every n n displaystyle n n this is written as lim n a n displaystyle lim _ n to infty a_ n infty similarly the sequence is said to tend to negative infinity if all but finitely many terms are less than any given lower bound this is written as lim n a n displaystyle lim _ n to infty a_ n infty in each case where lim n a n displaystyle lim _ n to infty a_ n infty or lim n a n displaystyle lim _ n to infty a_ n infty the sequence is said to have an infinite limit but it does not have a limit in the sense of being convergent to a real number since displaystyle infty and displaystyle infty are not real numbers metric space edit the discussion of sequences above is for sequences of real numbers the notion of limits can be defined for sequences valued in more abstract spaces such as metric spaces if m displaystyle m is a metric space with distance function d displaystyle d and a n n 0 displaystyle a_ n _ n geq 0 is a sequence in m displaystyle m then the limit when it exists of the sequence is an element a m displaystyle a in m such that given ε 0 displaystyle varepsilon 0 there exists an n displaystyle n such that for each n n displaystyle n n we have d a a n ε displaystyle d a a_ n varepsilon an equivalent statement is that a n a displaystyle a_ n rightarrow a if the sequence of real numbers d a a n 0 displaystyle d a a_ n rightarrow 0 example ℝ n edit an important example is the space of n displaystyle n dimensional real vectors with elements x x 1 x n displaystyle mathbf x x_ 1 cdots x_ n where each of the x i displaystyle x_ i are real an example of a suitable distance function is the euclidean distance defined by d x y x y i x i y i 2 displaystyle d mathbf x mathbf y mathbf x mathbf y sqrt sum _ i x_ i y_ i 2 the sequence of points x n n 0 displaystyle mathbf x _ n _ n geq 0 converges to x displaystyle mathbf x if the limit exists and x n x 0 displaystyle mathbf x _ n mathbf x rightarrow 0 topological space edit in some sense the most abstract space in which limits can be defined are topological spaces if x displaystyle x is a topological space with topology τ displaystyle tau and a n n 0 displaystyle a_ n _ n geq 0 is a sequence in x displaystyle x then the limit when it exists of the sequence is a point a x displaystyle a in x such that given a open neighborhood u τ displaystyle u in tau of a displaystyle a there exists an n displaystyle n such that for every n n displaystyle n n a n u displaystyle a_ n in u is satisfied in this case the limit if it exists may not be unique however it must be unique if x displaystyle x is a hausdorff space function space edit this section deals with the idea of limits of sequences of functions not to be confused with the idea of limits of functions discussed below the field of functional analysis partly seeks to identify useful notions of convergence on function spaces for example consider the space of functions from a generic set e displaystyle e to r displaystyle mathbb r given a sequence of functions f n n 0 displaystyle f_ n _ n 0 such that each is a function f n e r displaystyle f_ n e rightarrow mathbb r suppose that there exists a function such that for each x e displaystyle x in e f n x f x or equivalently lim n f n x f x displaystyle f_ n x rightarrow f x text or equivalently lim _ n rightarrow infty f_ n x f x then the sequence f n displaystyle f_ n is said to converge pointwise to f displaystyle f however such sequences can exhibit unexpected behavior for example it is possible to construct a sequence of continuous functions which has a discontinuous pointwise limit another notion of convergence is uniform convergence the uniform distance between two functions f g e r displaystyle f g e rightarrow mathbb r is the maximum difference between the two functions as the argument x e displaystyle x in e is varied that is d f g max x e f x g x displaystyle d f g max _ x in e f x g x then the sequence f n displaystyle f_ n is said to uniformly converge or have a uniform limit of f displaystyle f if f n f displaystyle f_ n rightarrow f with respect to this distance the uniform limit has nicer properties than the pointwise limit for example the uniform limit of a sequence of continuous functions is continuous many different notions of convergence can be defined on function spaces this is sometimes dependent on the regularity of the space prominent examples of function spaces with some notion of convergence are lp spaces and sobolev space in functions edit main article limit of a function a function f x for which the limit at infinity is l for any arbitrary distance ε there must be a value s such that the function stays within l ε for all x s suppose f is a real valued function and c is a real number intuitively speaking the expression lim x c f x l displaystyle lim _ x to c f x l means that f x can be made to be as close to l as desired by making x sufficiently close to c 10 in that case the above equation can be read as the limit of f of x as x approaches c is l formally the definition of the limit of f x displaystyle f x as x displaystyle x approaches c displaystyle c is given as follows the limit is a real number l displaystyle l so that given an arbitrary real number ε 0 displaystyle varepsilon 0 thought of as the error there is a δ 0 displaystyle delta 0 such that for any x displaystyle x satisfying 0 x c δ displaystyle 0 x c delta it holds that f x l ε displaystyle f x l varepsilon this is known as the ε δ definition of limit the inequality 0 x c displaystyle 0 x c is used to exclude c displaystyle c from the set of points under consideration but some authors do not include this in their definition of limits replacing 0 x c δ displaystyle 0 x c delta with simply x c δ displaystyle x c delta this replacement is equivalent to additionally requiring that f displaystyle f be continuous at c displaystyle c it can be proven that there is an equivalent definition which makes manifest the connection between limits of sequences and limits of functions 11 the equivalent definition is given as follows first observe that for every sequence x n displaystyle x_ n in the domain of f displaystyle f there is an associated sequence f x n displaystyle f x_ n the image of the sequence under f displaystyle f the limit is a real number l displaystyle l so that for all sequences x n c displaystyle x_ n rightarrow c the associated sequence f x n l displaystyle f x_ n rightarrow l one sided limit edit main article one sided limit it is possible to define the notion of having a left handed limit from below and a notion of a right handed limit from above these need not agree an example is given by the positive indicator function f r r displaystyle f mathbb r rightarrow mathbb r defined such that f x 0 displaystyle f x 0 if x 0 displaystyle x leq 0 and f x 1 displaystyle f x 1 if x 0 displaystyle x 0 at x 0 displaystyle x 0 the function has a left handed limit of 0 a right handed limit of 1 and its limit does not exist symbolically this can be stated as for this example lim x c f x 0 displaystyle lim _ x to c f x 0 and lim x c f x 1 displaystyle lim _ x to c f x 1 and from this it can be deduced lim x c f x displaystyle lim _ x to c f x does not exist because lim x c f x lim x c f x displaystyle lim _ x to c f x neq lim _ x to c f x infinity in limits of functions edit it is possible to define the notion of tending to infinity in the domain of f displaystyle f lim x f x l displaystyle lim _ x rightarrow infty f x l this could be considered equivalent to the limit as a reciprocal tends to 0 lim x 0 f 1 x l displaystyle lim _ x rightarrow 0 f 1 x l or it can be defined directly the limit of f displaystyle f as x displaystyle x tends to positive infinity is defined as a value l displaystyle l such that given any real ε 0 displaystyle varepsilon 0 there exists an m 0 displaystyle m 0 so that for all x m displaystyle x m f x l ε displaystyle f x l varepsilon the definition for sequences is equivalent as n displaystyle n rightarrow infty we have f x n l displaystyle f x_ n rightarrow l in these expressions the infinity is normally considered to be signed displaystyle infty or displaystyle infty and corresponds to a one sided limit of the reciprocal a two sided infinite limit can be defined but an author would explicitly write displaystyle pm infty to be clear it is also possible to define the notion of tending to infinity in the value of f displaystyle f lim x c f x displaystyle lim _ x rightarrow c f x infty again this could be defined in terms of a reciprocal lim x c 1 f x 0 displaystyle lim _ x rightarrow c frac 1 f x 0 or a direct definition can be given as follows given any real number m 0 displaystyle m 0 there is a δ 0 displaystyle delta 0 so that for 0 x c δ displaystyle 0 x c delta the absolute value of the function f x m displaystyle f x m a sequence can also have an infinite limit as n displaystyle n rightarrow infty the sequence f x n displaystyle f x_ n rightarrow infty this direct definition is easier to extend to one sided infinite limits while mathematicians do talk about functions approaching limits from above or from below there is not a standard mathematical notation for this as there is for one sided limits nonstandard analysis edit in non standard analysis which involves a hyperreal enlargement of the number system the limit of a sequence a n displaystyle a_ n can be expressed as the standard part of the value a h displaystyle a_ h of the natural extension of the sequence at an infinite hypernatural index n h thus lim n a n st a h displaystyle lim _ n to infty a_ n operatorname st a_ h here the standard part function st rounds off each finite hyperreal number to the nearest real number the difference between them is infinitesimal this formalizes the natural intuition that for very large values of the index the terms in the sequence are very close to the limit value of the sequence conversely the standard part of a hyperreal a a n displaystyle a a_ n represented in the ultrapower construction by a cauchy sequence a n displaystyle a_ n is simply the limit of that sequence st a lim n a n displaystyle operatorname st a lim _ n to infty a_ n in this sense taking the limit and taking the standard part are equivalent procedures limit sets edit limit set of a sequence edit let a n n 0 displaystyle a_ n _ n 0 be a sequence in a topological space x displaystyle x for concreteness x displaystyle x can be thought of as r displaystyle mathbb r but the definitions hold more generally the limit set is the set of points such that if there is a convergent subsequence a n k k 0 displaystyle a_ n_ k _ k 0 with a n k a displaystyle a_ n_ k rightarrow a then a displaystyle a belongs to the limit set in this context such an a displaystyle a is sometimes called a limit point a use of this notion is to characterize the long term behavior of oscillatory sequences for example consider the sequence a n 1 n displaystyle a_ n 1 n starting from n 1 the first few terms of this sequence are 1 1 1 1 displaystyle 1 1 1 1 cdots it can be checked that it is oscillatory so has no limit but has limit points 1 1...