In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input which may or may not be in the domain of the function. Formal definitions, first devised in the early 19th century, are given below. Informally, a function f assigns an output f(x) to every input x. We say that the function has a limit L at an input p, if f(x) gets closer and closer to L as x moves closer and closer to p. More specifically, the output value can be made arbitrarily close to L if the input to f is taken sufficiently close to p. On the other hand, if some inputs very close to p are taken to outputs that stay a fixed distance apart, then we say the limit does not exist. The notion of a limit has many applications in modern calculus. In particular, the many definitions of continuity employ the concept of limit: roughly, a function is continuous if all of its limits agree with the values of the function. The concept of limit also appears in the definition of the derivative: in the calculus of one variable, this is the limiting value of the slope of secant lines to the graph of a function.
History Although implicit in the development of calculus of the 17th and 18th centuries, the modern idea of the limit of a function goes back to Bernard Bolzano who, in 1817, introduced the basics of the epsilon-delta technique (see (ε, δ)-definition of limit below) to define continuous functions. However, his work was not known during his lifetime. Bruce Pourciau argues that Isaac Newton, in his 1687 Principia, demonstrates a more sophisticated understanding of limits than he is generally given credit for, including being the first to present an epsilon argument. In his 1821 book Cours d'analyse, Augustin-Louis Cauchy discussed variable quantities, infinitesimals and limits, and defined continuity of y = f ( x ) {\displaystyle y=f(x)} by saying that an infinitesimal change in x necessarily produces an infinitesimal change in y, while Grabiner claims that he used a rigorous epsilon-delta definition in proofs. In 1861, Karl Weierstrass first introduced the epsilon-delta definition of limit in the form it is usually written today. He also introduced the notations lim {\textstyle \lim } and lim x → x 0 . {\textstyle \textstyle \lim \limits _{x\to x_{0}}.\displaystyle }
The modern notation of placing the arrow below the limit symbol is due to G. H. Hardy, which is introduced in his book A Course of Pure Mathematics in 1908.
Functions of a single variable Informally, a function f ( x ) {\displaystyle f(x)} has limit L {\displaystyle L} as x {\displaystyle x} approaches a {\displaystyle a} if f ( x ) {\displaystyle f(x)} approximates L {\displaystyle L} for x {\displaystyle x} near a {\displaystyle a} . More precisely, the value of f ( x ) {\displaystyle f(x)} is within a given tolerance of L {\displaystyle L} , provided x {\displaystyle x} is within a corresponding tolerance of a {\displaystyle a} . These two tolerances are often denoted, respectively, by ε {\displaystyle \varepsilon } (the tolerance in the value of f ( x ) {\displaystyle f(x)} ) and δ {\displaystyle \delta } (the corresponding tolerance in x {\displaystyle x} ). The value of the function at x = a {\displaystyle x=a} is usually omitted from the approximation; for example, in many cases where limits are useful, the function has no value at x = a {\displaystyle x=a} (it is undefined there).
(ε, δ)-definition of limit
Suppose f : R → R {\displaystyle f:\mathbb {R} \rightarrow \mathbb {R} } is a function defined on the real line, and there are two real numbers p and L. One would say: "The limit of f of x, as x approaches p, exists, and it equals L". and write,
lim x → p f ( x ) = L , {\displaystyle \lim _{x\to p}f(x)=L,}
or alternatively, say "f(x) tends to L as x tends to p", and write,
f ( x ) → L as x → p , {\displaystyle f(x)\to L{\text{ as }}x\to p,}
if the following property holds: for every real ε > 0, there exists a real δ > 0 such that for all real x, 0 < |x − p| < δ implies |f(x) − L| < ε. Symbolically,
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