Limit of a function

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.

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 \(y=f(x)\) 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 \(\lim\) and \(\textstyle \lim\limits_{x \to x_0}. \displaystyle\)

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.

Informally, a function \(f(x)\) has limit \(L\) as \(x\) approaches \(a\) if \(f(x)\) approximates \(L\) for \(x\) near \(a\). More precisely, the value of \(f(x)\) is within a given tolerance of \(L\), provided \(x\) is within a corresponding tolerance of \(a\). These two tolerances are often denoted, respectively, by \(\varepsilon\) (the tolerance in the value of \(f(x)\)) and \(\delta\) (the corresponding tolerance in \(x\)). The value of the function at \(x=a\) is usually omitted from the approximation; for example, in many cases where limits are useful, the function has no value at \(x=a\) (it is undefined there).

Suppose \(f: \R \rightarrow \R\) 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, \[\lim_{x \to p} f(x) = L,\] or alternatively, say "f(x) tends to L as x tends to p", and write, \[f(x) \to L \text{ as } x \to p,\] 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, \[(\forall \varepsilon > 0 ) \, (\exists \delta > 0) \, (\forall x \in \R) \, (0 < |x - p| < \delta \implies |f(x) - L| < \varepsilon).\]

For example, one may say \[\lim_{x \to 2} (4x + 1) = 9\] because for every real ε > 0, we can take δ = ε/4, so that for all real x, if 0 < |x − 2| < δ, then |4x + 1 − 9| < ε.

A more general definition applies for functions defined on subsets of the real line. Let S be a subset of ⁠\(\R.\)⁠ Let \(f: S \to \R\) be a real-valued function. Let p be a point such that there exists some open interval (a, b) containing p with \((a,p)\cup (p,b) \subset S.\) It is then said that the limit of f as x approaches p is L, if:

Symbolically, \[(\forall \varepsilon > 0 ) \, (\exists \delta > 0) \, (\forall x \in (a, b)) \, (0 < |x - p| < \delta \implies |f(x) - L| < \varepsilon).\]

For example, one may say \[\lim_{x \to 1} \sqrt{x+3} = 2\] because for every real ε > 0, we can take δ = ε, so that for all real x ≥ −3, if 0 < |x − 1| < δ, then |f(x) − 2| < ε. In this example, S = [−3, ∞) contains open intervals around the point 1 (for example, the interval (0, 2)).

Here, note that the value of the limit does not depend on f being defined at p, nor on the value f(p), if it is defined. For example, let \(f: [0,1)\cup (1,2] \to \R, f(x) = \tfrac{2x^2 - x - 1}{x-1}.\) \[\lim_{x \to 1} f(x) = 3\] because for every ε > 0, we can take δ = ε/2, so that for all real x ≠ 1, if 0 < |x − 1| < δ, then |f(x) − 3| < ε. Note that here f(1) is undefined.

In fact, a limit can exist in \(\{p\in \R\, |\, \exists (a,b) \subset \R : \, p \in (a,b) \text{ and } (a,p)\cup (p, b) \subset S\},\) which equals \(\operatorname{int} S \cup \operatorname{iso} S^c,\) where int S is the interior of S, and iso S are the isolated points of the complement of S. In our previous example where \(S = [0,1) \cup (1,2],\) \(\operatorname{int} S = (0,1) \cup (1,2),\) \(\operatorname{iso} S^c = \{1\}.\) We see, specifically, this definition of limit allows a limit to exist at 1, but not at 0 or 2.

Condensed: the full section is in Wikipedia.

Alternatively, x may approach p from above (right) or below (left), in which case the limits may be written as

\[\lim_{x \to p^+}f(x) = L\]

\[\lim_{x \to p^-}f(x) = L\]

respectively. If these limits exist at p and are equal there, then this can be referred to as the limit of f(x) at p. If the one-sided limits exist at p, but are unequal, then there is no limit at p (i.e., the limit at p does not exist). If either one-sided limit does not exist at p, then the limit at p also does not exist.

A formal definition is as follows. The limit of f as x approaches p from above is L if:

For every ε > 0, there exists a δ > 0 such that whenever 0 < x − p < δ, we have |f(x) − L| < ε.

\[(\forall \varepsilon > 0 ) \, (\exists \delta > 0) \, (\forall x \in (a,b))\, (0 < x - p < \delta \implies |f(x) - L| < \varepsilon).\]

For every ε > 0, there exists a δ > 0 such that whenever 0 < p − x < δ, we have |f(x) − L| < ε.

Condensed: the full section is in Wikipedia.

Limits can also be defined by approaching from subsets of the domain.

In general: Let \(f : S \to \R\) be a real-valued function defined on some \(S \subseteq \R.\) Let p be a limit point of some \(T \subset S\), that is, p is the limit of some sequence of elements of T distinct from p. Then we say the limit of f, as x approaches p from values in T, is L, written \[\lim_{ {x \to p} \atop {x \in T} } f(x) = L\] if the following holds:

\[(\forall \varepsilon > 0 )\, (\exists \delta > 0) \,(\forall x \in T)\, (0 < |x - p| < \delta \implies |f(x) - L| < \varepsilon).\]

Note, T can be any subset of S, the domain of f. And the limit might depend on the selection of T. This generalization includes as special cases limits on an interval, as well as left-handed limits of real-valued functions (e.g., by taking T to be an open interval of the form (–∞, a)), and right-handed limits (e.g., by taking T to be an open interval of the form (a, ∞)). It also extends the notion of one-sided limits to the included endpoints of (half-)closed intervals, so the square root function \(f(x) = \sqrt x\) can have limit 0 as x approaches 0 from above: \[\lim_{ {x\to 0} \atop {x\in [0, \infty)} } \sqrt{x} = 0\] since for every ε > 0, we may take δ = ε such that for all x ≥ 0, if 0 < |x − 0| < δ, then |f(x) − 0| < ε.

This definition allows a limit to be defined at limit points of the domain S, if a suitable subset T which has the same limit point is chosen.

Notably, the previous two-sided definition works on \(\operatorname{int} S \cup \operatorname{iso} S^c,\) which is a subset of the limit points of S.

For example, let \(S = [0,1)\cup (1, 2].\) The previous two-sided definition would work at \(1 \in \operatorname{iso} S^c = \{1\},\) but it wouldn't work at 0 or 2, which are limit points of S.

The definition of limit given here does not depend on how (or whether) f is defined at p. Bartle refers to this as a deleted limit, because it excludes the value of f at p. The corresponding non-deleted limit does depend on the value of f at p, if p is in the domain of f. Let \(f : S \to \R\) be a real-valued function. The non-deleted limit of f, as x approaches p, is L if

\[(\forall \varepsilon > 0 )\, (\exists \delta > 0) \, (\forall x \in S)\, (|x - p| < \delta \implies |f(x) - L| < \varepsilon).\]

The definition is the same, except that the neighborhood |x − p| < δ now includes the point p, in contrast to the deleted neighborhood 0 < |x − p| < δ. This makes the definition of a non-deleted limit less general. One of the advantages of working with non-deleted limits is that they allow to state the theorem about limits of compositions without any constraints on the functions (other than the existence of their non-deleted limits).

Bartle notes that although by "limit" some authors do mean this non-deleted limit, deleted limits are the most popular.

Let \(f:S \to \R\) be a function defined on \(S \subseteq \R.\) The limit of f as x approaches infinity is L, denoted

\[\lim_{x \to \infty}f(x) = L,\]

means that:

\[(\forall \varepsilon > 0 )\, (\exists c > 0) \,(\forall x \in S) \,(x > c \implies |f(x) - L| < \varepsilon).\]

Similarly, the limit of f as x approaches minus infinity is L, denoted

\[\lim_{x \to -\infty}f(x) = L,\]

means that:

Condensed: the full section is in Wikipedia.

For a function whose values grow without bound, the function diverges and the usual limit does not exist. However, in this case one may introduce limits with infinite values.

Let \(f:S \to\mathbb{R}\) be a function defined on \(S\subseteq\mathbb{R}.\) The statement the limit of f as x approaches p is infinity, denoted

\[\lim_{x \to p} f(x) = \infty,\]

means that:

\[(\forall N > 0)\, (\exists \delta > 0)\, (\forall x \in S)\, (0 < | x-p | < \delta \implies f(x) > N) .\]

The statement the limit of f as x approaches p is minus infinity, denoted

\[\lim_{x \to p} f(x) = -\infty,\]

• a neighborhood of −∞ is defined to contain an interval [−∞, c) for some ⁠\(c \in \R,\)⁠
• a neighborhood of ∞ is defined to contain an interval (c, ∞] where ⁠\(c \in \R,\)⁠ and
• a neighborhood of ⁠\(a \in \R\)⁠ is defined in the normal way metric space ⁠\(\R.\)⁠

Condensed: the full section is in Wikipedia.

Many authors allow for the projectively extended real line to be used as a way to include infinite values as well as extended real line. With this notation, the extended real line is given as ⁠\(\R \cup \{-\infty, +\infty\}\)⁠ and the projectively extended real line is ⁠\(\R \cup \{\infty\}\)⁠ where a neighborhood of ∞ is a set of the form \(\{x: |x| > c\}.\) The advantage is that one only needs three definitions for limits (left, right, and central) to cover all the cases. As presented above, for a completely rigorous account, we would need to consider 15 separate cases for each combination of infinities (five directions: −∞, left, central, right, and +∞; three bounds: −∞, finite, or +∞). There are also noteworthy pitfalls. For example, when working with the extended real line, \(x^{-1}\) does not possess a central limit (which is normal):

\[\lim_{x \to 0^{+}}{1\over x} = +\infty, \quad \lim_{x \to 0^{-}}{1\over x} = -\infty.\]

In contrast, when working with the projective real line, infinities (much like 0) are unsigned, so, the central limit does exist in that context:

\[\lim_{x \to 0^{+}}{1\over x} = \lim_{x \to 0^{-}}{1\over x} = \lim_{x \to 0}{1\over x} = \infty.\]

In fact there are a plethora of conflicting formal systems in use. In certain applications of numerical differentiation and integration, it is, for example, convenient to have signed zeroes. A simple reason has to do with the converse of \(\lim_{x \to 0^{-}}{x^{-1}} = -\infty,\) namely, it is convenient for \(\lim_{x \to -\infty}{x^{-1}} = -0\) to be considered true. Such zeroes can be seen as an approximation to infinitesimals.

There are three basic rules for evaluating limits at infinity for a rational function \(f(x) = \tfrac{p(x)}{q(x)}\) (where p and q are polynomials):

• If the degree of p is greater than the degree of q, then the limit is positive or negative infinity depending on the signs of the leading coefficients;
• If the degree of p and q are equal, the limit is the leading coefficient of p divided by the leading coefficient of q;
• If the degree of p is less than the degree of q, the limit is 0.

If the limit at infinity exists, it represents a horizontal asymptote at y = L. Polynomials do not have horizontal asymptotes; such asymptotes may however occur with rational functions.

By noting that |x − p| represents a distance, the definition of a limit can be extended to functions of more than one variable. In the case of a function \(f : S \times T \to \R\) defined on \(S \times T \subseteq \R^2,\) we defined the limit as follows: the limit of f as (x, y) approaches (p, q) is L, written

\[\lim_{(x,y) \to (p, q)} f(x, y) = L\]

if the following condition holds:

For every ε > 0, there exists a δ > 0 such that for all x in S and y in T, whenever \(0 < \sqrt{(x-p)^2 + (y-q)^2} < \delta,\) we have |f(x, y) − L| < ε,

or formally: \[(\forall \varepsilon > 0)\, (\exists \delta > 0)\, (\forall x \in S) \, (\forall y \in T)\, (0 < \sqrt{(x-p)^2 + (y-q)^2} < \delta \implies |f(x, y) - L| < \varepsilon)).\]

Here \(\sqrt{(x-p)^2 + (y-q)^2}\) is the Euclidean distance between (x, y) and (p, q). (This can in fact be replaced by any norm ||(x, y) − (p, q)||, and be extended to any number of variables.)

For example, we may say \[\lim_{(x,y) \to (0, 0)} \frac{x^4}{x^2+y^2} = 0\] because for every ε > 0, we can take \(\delta = \sqrt \varepsilon\) such that for all real x ≠ 0 and real y ≠ 0, if \(0 < \sqrt{(x-0)^2 + (y-0)^2} < \delta,\) then |f(x, y) − 0| < ε.

Similar to the case in single variable, the value of f at (p, q) does not matter in this definition of limit.

Condensed: the full section is in Wikipedia.

Although less commonly used, there is another type of limit for a multivariable function, known as the multiple limit. For a two-variable function, this is the double limit. Let \(f : S \times T \to \R\) be defined on \(S \times T \subseteq \R^2,\) we say the double limit of f as x approaches p and y approaches q is L, written

\[\lim_{ {x \to p} \atop {y \to q} } f(x, y) = L\]

if the following condition holds:

\[(\forall \varepsilon > 0)\, (\exists \delta > 0)\, (\forall x \in S) \, (\forall y \in T)\, ( (0 < |x-p| < \delta) \land (0 < |y-q| < \delta) \implies |f(x, y) - L| < \varepsilon) .\]

For such a double limit to exist, this definition requires the value of f approaches L along every possible path approaching (p, q), excluding the two lines x = p and y = q. As a result, the multiple limit is a weaker notion than the ordinary limit: if the ordinary limit exists and equals L, then the multiple limit exists and also equals L. The converse is not true: the existence of the multiple limits does not imply the existence of the ordinary limit. Consider the example \[f(x,y) = \begin{cases} 1 \quad \text{for} \quad xy \ne 0 \\ 0 \quad \text{for} \quad xy = 0 \end{cases}\] where \[\lim_{ {x \to 0} \atop {y \to 0} } f(x, y) = 1\] but \[\lim_{(x, y) \to (0, 0)} f(x, y)\] does not exist.

If the domain of f is restricted to \((S\setminus\{p\}) \times (T\setminus\{q\}),\) then the two definitions of limits coincide.

The concept of multiple limit can extend to the limit at infinity, in a way similar to that of a single variable function. For \(f : S \times T \to \R,\) we say the double limit of f as x and y approaches infinity is L, written \[\lim_{ {x \to \infty} \atop {y \to \infty} } f(x, y) = L\]

if the following condition holds:

\[(\forall \varepsilon > 0)\, (\exists c> 0)\, (\forall x \in S) \, (\forall y \in T)\, ( (x > c) \land (y > c) \implies |f(x, y) - L| < \varepsilon) .\]

We say the double limit of f as x and y approaches minus infinity is L, written \[\lim_{ {x \to -\infty} \atop {y \to -\infty} } f(x, y) = L\]

if the following condition holds:

\[(\forall \varepsilon > 0)\, (\exists c> 0)\, (\forall x \in S) \, (\forall y \in T)\, ( (x < -c) \land (y < -c) \implies |f(x, y) - L| < \varepsilon) .\]