Limit of a function

Suppose ${\displaystyle f:\mathbb {R} \rightarrow \mathbb {R} }$ is a function defined on the real line, and there are two real numbers p and L. One would say that the limit of f, as x approaches p, is L and written

${\displaystyle \lim _{x\to p}f(x)=L}$,

or alternatively, say f(x) tends to L as x tends to p, and written:

${\displaystyle f(x)\to L\;\;{\text{as}}\;\;x\to p}$,

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.[6]

Or, symbolically:

${\displaystyle (\forall \varepsilon >0)\,(\exists \delta >0)\,(\forall x\in \mathbb {R} )\,(0<|x-p|<\delta \implies |f(x)-L|<\varepsilon )}$.

For example, we may say

${\displaystyle \lim _{x\to 2}4x+1=9}$

because for every real ε > 0, we can take δ = ε/4, so that for all real x, if 0 < |x − p| < δ, then |f(x) − L| < ε.

A more general definition applies for functions defined on subsets of the real line. Let (a, b) be an open interval in ${\displaystyle \mathbb {R} }$, and a number p in (a, b). Let ${\displaystyle f:S\to \mathbb {R} }$ be a real-valued function defined on S — a set contains all of (a, b), except possibly at p itself. It is then said that the limit of f as x approaches p is L, if:

For every real ε > 0, there exists a real δ > 0 such that for all x ∈ (a, b), 0 < |x − p| < δ implies that |f(x) − L| < ε.

Or, symbolically:

${\displaystyle (\forall \varepsilon >0)\,(\exists \delta >0)\,(\forall x\in (a,b))\,(0<|x-p|<\delta \implies |f(x)-L|<\varepsilon )}$.

For example, we may say

${\displaystyle \lim _{x\to 1}{\sqrt {x+3}}=2}$

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,

${\displaystyle \lim _{x\to 1}{\frac {2x^{2}-x-1}{x-1}}=3}$

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.

The letters ε and δ can be understood as "error" and "distance". In fact, Cauchy used ε as an abbreviation for "error" in some of his work,[2] though in his definition of continuity, he used an infinitesimal ${\displaystyle \alpha }$ rather than either ε or δ (see Cours d'Analyse). In these terms, the error (ε) in the measurement of the value at the limit can be made as small as desired, by reducing the distance (δ) to the limit point. As discussed below, this definition also works for functions in a more general context. The idea that δ and ε represent distances helps suggest these generalizations.

The limit as: x → x₀⁺ ≠ x → x₀⁻. Therefore, the limit as x → x₀ does not exist.

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

${\displaystyle \lim _{x\to p^{+}}f(x)=L}$

or

${\displaystyle \lim _{x\to p^{-}}f(x)=L}$

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.[7] 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.

${\displaystyle (\forall \varepsilon >0)\,(\exists \delta >0)\,(\forall x\in (a,b))\,(0<x-p<\delta \implies |f(x)-L|<\varepsilon )}$.

The limit of f as x approaches p from below is L if:

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

${\displaystyle (\forall \varepsilon >0)\,(\exists \delta >0)\,(\forall x\in (a,b))\,(0<p-x<\delta \implies |f(x)-L|<\varepsilon )}$.

If the limit does not exist, then the oscillation of f at p is non-zero.

Apart from open intervals, limits can be defined for functions on arbitrary subsets of R, as follows (Bartle & Sherbert 2000) harv error: no target: CITEREFBartleSherbert2000 (help): let ${\displaystyle f:S\to \mathbb {R} }$ be a real-valued function defined on arbitrary ${\displaystyle S\subseteq \mathbb {R} }$. Let p be a limit point of S—that is, p is the limit of some sequence of elements of S distinct from p. Then we say the limit of f, as x approaches p from values in S, is L, written

${\displaystyle \lim _{\begin{smallmatrix}x\to p\\x\in S\end{smallmatrix}}f(x)=L}$

if the following holds:

For every ε > 0, there exists a δ > 0 such that for all x ∈ S, 0 < |x − p| < δ implies that |f(x) − L| < ε.

${\displaystyle (\forall \varepsilon >0)\,(\exists \delta >0)\,(\forall x\in S)\,(0<|x-p|<\delta \implies |f(x)-L|<\varepsilon )}$.

The condition that f be defined on S is that S be a subset of the domain of f. This generalization includes as special cases limits on an interval, as well as left-handed limits of real-valued functions (e.g., by taking S to be an open interval of the form ${\displaystyle (-\infty ,a)}$), and right-handed limits (e.g., by taking S to be an open interval of the form ${\displaystyle (a,\infty )}$). It also extends the notion of one-sided limits to the included endpoints of (half-)closed intervals, so the square root function f(x) = √x can have limit 0 as x approaches 0 from above:

${\displaystyle \lim _{\begin{smallmatrix}x\to 0\\x\in [0,\infty )\end{smallmatrix}}{\sqrt {x}}=0}$

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.

The definition of limit given here does not depend on how (or whether) f is defined at p. Bartle (1967) 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 ${\displaystyle f:S\to \mathbb {R} }$ be a real-valued function. The non-deleted limit of f, as x approaches p, is L if

For every ε > 0, there exists a δ > 0 such that for all x ∈ S, |x − p| < δ implies |f(x) − L| < ε

${\displaystyle (\forall \varepsilon >0)\,(\exists \delta >0)\,(\forall x\in S)\,(|x-p|<\delta \implies |f(x)-L|<\varepsilon )}$.

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) (Hubbard (2015)).

Bartle (1967) notes that although by "limit" some authors do mean this non-deleted limit, deleted limits are the most popular. For example, Apostol (1974), Courant (1924), Hardy (1921), Rudin (1964), Whittaker & Watson (1902) harvtxt error: no target: CITEREFWhittakerWatson1902 (help) all take "limit" to mean the deleted limit.

The function without a limit, at an essential discontinuity

The function

${\displaystyle f(x)={\begin{cases}\sin {\frac {5}{x-1}}&{\text{ for }}x<1\\0&{\text{ for }}x=1\\{\frac {0.1}{x-1}}&{\text{ for }}x>1\end{cases}}}$

has no limit at ${\displaystyle x_{0}=1}$ (the left-hand limit does not exist due to the oscillatory nature of the sine function, and the right-hand limit does not exist due to the asymptotic behaviour of the reciprocal function), but has a limit at every other x-coordinate.

The function

${\displaystyle f(x)={\begin{cases}1&x{\text{ rational }}\\0&x{\text{ irrational }}\end{cases}}}$

(a.k.a., the Dirichlet function) has no limit at any x-coordinate.

The function

${\displaystyle f(x)={\begin{cases}1&{\text{ for }}x<0\\2&{\text{ for }}x\geq 0\end{cases}}}$

has a limit at every non-zero x-coordinate (the limit equals 1 for negative x and equals 2 for positive x). The limit at x = 0 does not exist (the left-hand limit equals 1, whereas the right-hand limit equals 2).

The functions

${\displaystyle f(x)={\begin{cases}x&x{\text{ rational }}\\0&x{\text{ irrational }}\end{cases}}}$

and

${\displaystyle f(x)={\begin{cases}|x|&x{\text{ rational }}\\0&x{\text{ irrational }}\end{cases}}}$

both have a limit at x = 0 and it equals 0.

The function

${\displaystyle f(x)={\begin{cases}\sin x&x{\text{ irrational }}\\1&x{\text{ rational }}\end{cases}}}$

has a limit at any x-coordinate of the form ${\displaystyle {\frac {\pi }{2}}+2n\pi }$, where n is any integer.

The limit of this function at infinity exists.

Let ${\displaystyle f:S\to \mathbb {R} }$ be a function defined on ${\displaystyle S\subseteq \mathbb {R} }$. The limit of f as x approaches infinity is L, denoted

${\displaystyle \lim _{x\to \infty }f(x)=L}$,

means that:

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

${\displaystyle (\forall \varepsilon >0)\,(\exists c>0)\,(\forall x\in S)\,(x>c\implies |f(x)-L|<\varepsilon )}$.

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

${\displaystyle \lim _{x\to -\infty }f(x)=L}$,

means that:

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

${\displaystyle (\forall \varepsilon >0)\,(\exists c>0)\,(\forall x\in S)\,(x<-c\implies |f(x)-L|<\varepsilon )}$.

For example,

${\displaystyle \lim _{x\to \infty }\left(-{\frac {3\sin x}{x}}+4\right)=4}$

because for every ε > 0, we can take c = 3/ε such that for all real x, if x > c, then |f(x) − 4| < ε.

Another example is that

${\displaystyle \lim _{x\to -\infty }e^{x}=0}$

because for every ε > 0, we can take c = max{1, −ln(ε)} such that for all real x, if x < −c, then |f(x) − 0| < ε.

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

${\displaystyle \lim _{x\to p}f(x)=\infty ,}$

means that:

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

${\displaystyle (\forall N>0)\,(\exists \delta >0)\,(\forall x\in S)\,(0<|x-p|<\delta \implies f(x)>N)}$.

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

${\displaystyle \lim _{x\to p}f(x)=-\infty ,}$

means that:

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

${\displaystyle (\forall N>0)\,(\exists \delta >0)\,(\forall x\in S)\,(0<|x-p|<\delta \implies f(x)<-N)}$.

For example,

${\displaystyle \lim _{x\to 1}{\frac {1}{(x-1)^{2}}}=\infty }$

because for every N > 0, we can take δ = 1/√N such that for all real x > 0, if 0 < x − 1 < δ, then f(x) > N.

These ideas can be combined in a natural way to produce definitions for different combinations, such as

${\displaystyle \lim _{x\to \infty }f(x)=\infty }$, or ${\displaystyle \lim _{x\to p^{+}}f(x)=-\infty }$.

For example,

${\displaystyle \lim _{x\to 0^{+}}\ln x=-\infty }$

because for every N > 0, we can take δ = e^−N such that for all real x > 0, if 0 < x − 0 < δ, then f(x) < −N.

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

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

In this case, R is a topological space and any function of the form f: X → Y with X, Y⊆ R is subject to the topological definition of a limit. Note that with this topological definition, it is easy to define infinite limits at finite points, which have not been defined above in the metric sense.

Many authors[8] 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 ∪ {−∞, +∞} and the projectively extended real line is R ∪ {∞} 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, ${\displaystyle x^{-1}}$ does not possess a central limit (which is normal):

${\displaystyle \lim _{x\to 0^{+}}{1 \over x}=+\infty ,\lim _{x\to 0^{-}}{1 \over x}=-\infty }$.

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

${\displaystyle \lim _{x\to 0^{+}}{1 \over x}=\lim _{x\to 0^{-}}{1 \over x}=\lim _{x\to 0}{1 \over x}=\infty }$.

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 ${\displaystyle \lim _{x\to 0^{-}}{x^{-1}}=-\infty }$, namely, it is convenient for ${\displaystyle \lim _{x\to -\infty }{x^{-1}}=-0}$ to be considered true. Such zeroes can be seen as an approximation to infinitesimals.

Horizontal asymptote about y = 4

There are three basic rules for evaluating limits at infinity for a rational function f(x) = 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 ${\displaystyle f:S\times T\to \mathbb {R} }$ defined on ${\displaystyle S\times T\subseteq \mathbb {R} ^{2}}$, we defined the limit as follows: the limit of f as (x, y) approaches (p, q) is L, written

${\displaystyle \lim _{(x,y)\to (p,q)}f(x,y)=L}$

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 < √(x−p)² + (y−q)² < δ, we have |f(x, y) − L| < ε.[9]

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

Here √(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

${\displaystyle \lim _{(x,y)\to (0,0)}{\frac {x^{4}}{x^{2}+y^{2}}}=0}$

because for every ε > 0, we can take δ = √ε such that for all real x ≠ 0 and real y ≠ 0, if 0 < √(x−0)² + (y−0)² < δ, 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.

For such a multivariable limit to exist, this definition requires the value of f approaches L along every possible path approaching (p, q).[10] In the above example, the function

${\displaystyle f(x,y)={\frac {x^{4}}{x^{2}+y^{2}}}}$

satisfies this condition. This can be seen by considering the polar coordinates (x, y) = (r cos(θ), r sin(θ)) → (0, 0), which gives

${\displaystyle \lim _{r\to 0}f(r\cos \theta ,r\sin \theta )=\lim _{r\to 0}{\frac {r^{4}\cos ^{4}\theta }{r^{2}}}=\lim _{r\to 0}r^{2}\cos ^{4}\theta }$.

Here θ = θ(r) is a function of r which controls the shape of the path along which f is approaching (p, q). Since cos(θ) is bounded between [−1, 1], by the sandwich theorem, this limit tends to 0.

In contrast, the function

${\displaystyle f(x,y)={\frac {xy}{x^{2}+y^{2}}}}$

does not have a limit at (0, 0). Taking the path (x, y) = (t, 0) → (0, 0), we obtain

${\displaystyle \lim _{t\to 0}f(t,0)=\lim _{t\to 0}{\frac {0}{t^{2}}}=0}$,

while taking the path (x, y) = (t, t) → (0, 0), we obtain

${\displaystyle \lim _{t\to 0}f(t,t)=\lim _{t\to 0}{\frac {t^{2}}{t^{2}+t^{2}}}={\frac {1}{2}}}$.

Since the two values do not agree, f does not tend to a single value as (x, y) approaches (0, 0).

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.[11] Let ${\displaystyle f:S\times T\to \mathbb {R} }$ be defined on ${\displaystyle S\times T\subseteq \mathbb {R} ^{2}}$, we say the double limit of f as x approaches p and y approaches q is L, written

${\displaystyle \lim _{\begin{smallmatrix}x\to p\\y\to q\end{smallmatrix}}f(x,y)=L}$

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 < |x − p| < δ and 0 < |y−q| < δ, we have |f(x, y) − L| < ε.[11]

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

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. Note that the converse is not true: the existence of the multiple limits does not imply the existence of the ordinary limit. Consider the example

${\displaystyle f(x,y)={\begin{cases}1\quad {\text{for}}\quad xy\neq 0\\0\quad {\text{for}}\quad xy=0\end{cases}}}$

where

${\displaystyle \lim _{\begin{smallmatrix}x\to 0\\y\to 0\end{smallmatrix}}f(x,y)=1}$

but

${\displaystyle \lim _{(x,y)\to (0,0)}f(x,y)}$ does not exists.

If the domain of f is restricted to ${\displaystyle (S\setminus \{p\})\times (T\setminus \{q\})}$, then the two definitions of limits coincide.[11]

The concept of multiple limit can extend to the limit at infinity, in a way similar to that of a single variable function. For ${\displaystyle f:S\times T\to \mathbb {R} }$, we say the double limit of f as x and y approaches infinity is L, written

${\displaystyle \lim _{\begin{smallmatrix}x\to \infty \\y\to \infty \end{smallmatrix}}f(x,y)=L}$

if the following condition holds:

For every ε > 0, there exists a c > 0 such that for all x in S and y, whenever x > c and y > c, we have |f(x, y) − L| < ε.

${\displaystyle (\forall \varepsilon >0)\,(\exists c>0)\,(\forall x\in S)\,(\forall y\in T)\,((x>c)\land (y>c)\implies |f(x,y)-L|<\varepsilon )}$.

We say the double limit of f as x and y approaches minus infinity is L, written

${\displaystyle \lim _{\begin{smallmatrix}x\to -\infty \\y\to -\infty \end{smallmatrix}}f(x,y)=L}$

if the following condition holds:

For every ε > 0, there exists a c > 0 such that x in S and y in T, whenever x < −c and y < −c, we have |f(x, y) − L| < ε.

${\displaystyle (\forall \varepsilon >0)\,(\exists c>0)\,(\forall x\in S)\,(\forall y\in T)\,((x<-c)\land (y<-c)\implies |f(x,y)-L|<\varepsilon )}$.

Let ${\displaystyle f:S\times T\to \mathbb {R} }$. Instead of taking limit as (x, y) → (p, q), we may consider taking the limit of just one variable, say, x → p, to obtain a single-variable function of y, namely ${\displaystyle g:T\to \mathbb {R} }$. In fact, this limiting process can be done in two distinct ways. The first one is called pointwise limit. We say the pointwise limit of f as x approaches p is g, denoted

${\displaystyle \lim _{x\to p}f(x,y)=g(y)}$, or

${\displaystyle \lim _{x\to p}f(x,y)=g(y)\;\;{\text{pointwise}}}$.

Alternatively, we may say f tends to g pointwise as x approaches p, denoted

${\displaystyle f(x,y)\to g(y)\;\;{\text{as}}\;\;x\to p}$, or

${\displaystyle f(x,y)\to g(y)\;\;{\text{pointwise}}\;\;{\text{as}}\;\;x\to p}$.

This limit exists if the following holds:

For every ε > 0 and every fixed y in T, there exists a δ(ε, y) > 0 such that for all x in S, whenever 0 < |x − p| < δ, we have |f(x, y) − g(y)| < ε.[12]

${\displaystyle (\forall \varepsilon >0)\,(\forall y\in T)\,(\exists \delta >0)\,(\forall x\in S)\,(0<|x-p|<\delta \implies |f(x,y)-g(y)|<\varepsilon )}$.

Here, δ = δ(ε, y) is a function of both ε and y. Each δ is chosen for a specific point of y. Hence we say the limit is pointwise in y. For example,

${\displaystyle f(x,y)={\frac {x}{\cos y}}}$

has a pointwise limit of constant zero function

${\displaystyle \lim _{x\to 0}f(x,y)=0(y)\;\;{\text{pointwise}}}$

because for every fixed y, the limit is clearly 0. Note that this argument fails if y is not fixed: if y is very close to π/2, the value of the fraction may deviate from 0.

This leads to another definition of limit, namely the uniform limit. We say the uniform limit of f on T as x approaches p is g, denoted

${\displaystyle {\underset {\begin{smallmatrix}x\to p\\y\in T\end{smallmatrix}}{\mathrm {unif} \lim \;}}f(x,y)=g(y)}$, or

${\displaystyle \lim _{x\to p}f(x,y)=g(y)\;\;{\text{uniformly on}}\;T}$.

Alternatively, we may say f tends to g uniformly on T as x approaches p, denoted

${\displaystyle f(x,y)\rightrightarrows g(y)\;{\text{on}}\;T\;\;{\text{as}}\;\;x\to p}$, or

${\displaystyle f(x,y)\to g(y)\;\;{\text{uniformly on}}\;T\;\;{\text{as}}\;\;x\to p}$.

This limit exists if the following holds:

For every ε > 0, there exists a δ(ε) > 0 such that for all x in S and y in T, whenever 0 < |x − p| < δ, we have |f(x, y) − g(y)| < ε.[12]

${\displaystyle (\forall \varepsilon >0)\,(\exists \delta >0)\,(\forall x\in S)\,(\forall y\in T)\,(0<|x-p|<\delta \implies |f(x,y)-g(y)|<\varepsilon )}$.

Here, δ = δ(ε) is a function of only ε but not y. In other words, δ is uniformly applicable to all y in T. Hence we say the limit is uniform in y. For example,

${\displaystyle f(x,y)=x\cos y}$

has a uniform limit of constant zero function

${\displaystyle \lim _{x\to 0}f(x,y)=0(y)\;\;{\text{ uniformly on}}\;\mathbb {R} }$

because for all real y, cos(y) is bounded between [−1, 1]. Hence no matter how y behaves, we may use the sandwich theorem to show that the limit is 0.

Let ${\displaystyle f:S\times T\to \mathbb {R} }$. We may consider taking the limit of just one variable, say, x → p, to obtain a single-variable function of y, namely ${\displaystyle g:T\to \mathbb {R} }$, and then take limit in the other variable, namely y → q, to get a number ${\displaystyle L}$. Symbolically,

${\displaystyle \lim _{y\to q}\lim _{x\to p}f(x,y)=\lim _{y\to q}g(y)=L}$.

This limit is known as iterated limit of the multivariable function.[13] Note that the order of taking limits may affect the result, i.e.,

${\displaystyle \lim _{y\to q}\lim _{x\to p}f(x,y)\neq \lim _{x\to p}\lim _{y\to q}f(x,y)}$ in general.

A sufficient condition of equality is given by the Moore-Osgood theorem, which requires the limit ${\displaystyle \lim _{x\to p}f(x,y)=g(y)}$ to be uniform on T.[14]

The limit in Euclidean space is a direct generalization of limits to vector-valued functions. For example, we may consider a function ${\displaystyle f:S\times T\to \mathbb {R} ^{3}}$ such that

${\displaystyle f(x,y)=(f_{1}(x,y),f_{2}(x,y),f_{3}(x,y))}$.

Then, under the usual Euclidean metric,

${\displaystyle \lim _{(x,y)\to (p,q)}f(x,y)=(L_{1},L_{2},L_{3})}$

if the following holds:

For every ε > 0, there exists a δ > 0 such that for all x in S and y in T, 0 < √(x−p)² + (y−q)² < δ implies √(f₁−L₁)² + (f₂−L₂)² + (f₃−L₃)² < ε.[16]

${\displaystyle (\forall \varepsilon >0)\,(\exists \delta >0)\,(\forall x\in S)\,(\forall y\in T)\,(0<{\sqrt {(x-p)^{2}+(y-q)^{2}}}<\delta \implies {\sqrt {(f_{1}-L_{1})^{2}+(f_{2}-L_{2})^{2}+(f_{3}-L_{3})^{2}}}<\varepsilon )}$.

In this example, the function concerned are finite-dimension vector-valued function. In this case, the limit theorem for vector-valued function states that if the limit of each component exists, then the limit of a vector-valued function equals the vector with each component taken the limit:[16]

${\displaystyle \lim _{(x,y)\to (p,q)}\left(f_{1}(x,y),f_{2}(x,y),f_{3}(x,y)\right)=\left(\lim _{(x,y)\to (p,q)}f_{1}(x,y),\lim _{(x,y)\to (p,q)}f_{2}(x,y),\lim _{(x,y)\to (p,q)}f_{3}(x,y)\right)}$.

One might also want to consider spaces other than Euclidean space. An example would be the Manhattan space. Consider ${\displaystyle f:S\to \mathbb {R} ^{2}}$ such that

${\displaystyle f(x)=(f_{1}(x),f_{2}(x))}$.

Then, under the Manhattan metric,

${\displaystyle \lim _{x\to p}f(x)=(L_{1},L_{2})}$

if the following holds:

For every ε > 0, there exists a δ > 0 such that for all x in S, 0 < |x − p| < δ implies |f₁−L₁| + |f₂−L₂| < ε.

${\displaystyle (\forall \varepsilon >0)\,(\exists \delta >0)\,(\forall x\in S)\,(0<|x-p|<\delta \implies |f_{1}-L_{1}|+|f_{2}-L_{2}|<\varepsilon )}$.

Since this is also a finite-dimension vector-valued function, the limit theorem stated above also applies.[17]

Finally, we will discuss the limit in function space, which has infinite dimensions. Consider a function f(x, y) in the function space ${\displaystyle S\times T\to \mathbb {R} }$. We want to find out as x approaches p, how f(x, y) will tend to another function g(y), which is in the function space ${\displaystyle T\to \mathbb {R} }$. The "closeness" in this function space may be measured under the uniform metric.[18] Then, we will say the uniform limit of f on T as x approaches p is g and write

${\displaystyle {\underset {\begin{smallmatrix}x\to p\\y\in T\end{smallmatrix}}{\mathrm {unif} \lim \;}}f(x,y)=g(y)}$, or

${\displaystyle \lim _{x\to p}f(x,y)=g(y)\;\;{\text{uniformly on}}\;T}$,

if the following holds:

For every ε > 0, there exists a δ > 0 such that for all x in S, 0 < |x − p| < δ implies sup_y∈T |f(x,y) − g(y)| < ε.

${\displaystyle (\forall \varepsilon >0)\,(\exists \delta >0)\,(\forall x\in S)\,(0<|x-p|<\delta \implies \sup _{y\in T}|f(x,y)-g(y)|<\varepsilon )}$.

In fact, one can see that this definition is equivalent to that of the uniform limit of a multivariable function introduced in the previous section.

For functions on the real line, one way to define the limit of a function is in terms of the limit of sequences. (This definition is usually attributed to Eduard Heine.) In this setting:

${\displaystyle \lim _{x\to a}f(x)=L}$

if, and only if, for all sequences ${\displaystyle x_{n}}$ (with ${\displaystyle x_{n}}$ not equal to a for all n) converging to ${\displaystyle a}$ the sequence ${\displaystyle f(x_{n})}$ converges to ${\displaystyle L}$. It was shown by Sierpiński in 1916 that proving the equivalence of this definition and the definition above, requires and is equivalent to a weak form of the axiom of choice. Note that defining what it means for a sequence ${\displaystyle x_{n}}$ to converge to ${\displaystyle a}$ requires the epsilon, delta method.

Similarly as it was the case of Weierstrass's definition, a more general Heine definition applies to functions defined on subsets of the real line. Let f be a real-valued function with the domain Dm(f). Let a be the limit of a sequence of elements of Dm(f) \ {a}. Then the limit (in this sense) of f is L as x approaches p if for every sequence ${\displaystyle x_{n}}$ ∈ Dm(f) \ {a} (so that for all n, ${\displaystyle x_{n}}$ is not equal to a) that converges to a, the sequence ${\displaystyle f(x_{n})}$ converges to ${\displaystyle L}$. This is the same as the definition of a sequential limit in the preceding section obtained by regarding the subset Dm(f) of R as a metric space with the induced metric.

In non-standard calculus the limit of a function is defined by:

${\displaystyle \lim _{x\to a}f(x)=L}$

if and only if for all ${\displaystyle x\in \mathbb {R} ^{*}}$, ${\displaystyle f^{*}(x)-L}$ is infinitesimal whenever ${\displaystyle x-a}$ is infinitesimal. Here ${\displaystyle \mathbb {R} ^{*}}$ are the hyperreal numbers and ${\displaystyle f^{*}}$ is the natural extension of f to the non-standard real numbers. Keisler proved that such a hyperreal definition of limit reduces the quantifier complexity by two quantifiers.[19] On the other hand, Hrbacek writes that for the definitions to be valid for all hyperreal numbers they must implicitly be grounded in the ε-δ method, and claims that, from the pedagogical point of view, the hope that non-standard calculus could be done without ε-δ methods cannot be realized in full.[20] Bŀaszczyk et al. detail the usefulness of microcontinuity in developing a transparent definition of uniform continuity, and characterize Hrbacek's criticism as a "dubious lament".[21]

At the 1908 international congress of mathematics F. Riesz introduced an alternate way defining limits and continuity in concept called "nearness".[22] A point ${\displaystyle x}$ is defined to be near a set ${\displaystyle A\subseteq \mathbb {R} }$ if for every ${\displaystyle r>0}$ there is a point ${\displaystyle a\in A}$ so that ${\displaystyle |x-a|<r}$. In this setting the

${\displaystyle \lim _{x\to a}f(x)=L}$

if and only if for all ${\displaystyle A\subseteq \mathbb {R} }$, ${\displaystyle L}$ is near ${\displaystyle f(A)}$ whenever ${\displaystyle a}$ is near ${\displaystyle A}$. Here ${\displaystyle f(A)}$ is the set ${\displaystyle \{f(x)|x\in A\}}$. This definition can also be extended to metric and topological spaces.

In general, from knowing that

${\displaystyle \lim _{y\to b}f(y)=c}$ and ${\displaystyle \lim _{x\to a}g(x)=b}$,

it does not follow that ${\displaystyle \lim _{x\to a}f(g(x))=c}$. However, this "chain rule" does hold if one of the following additional conditions holds:

• f(b) = c (that is, f is continuous at b), or
• g does not take the value b near a (that is, there exists a ${\displaystyle \delta >0}$ such that if ${\displaystyle 0<|x-a|<\delta }$ then ${\displaystyle |g(x)-b|>0}$).

As an example of this phenomenon, consider the following function that violates both additional restrictions:

${\displaystyle f(x)=g(x)={\begin{cases}0&{\text{if }}x\neq 0\\1&{\text{if }}x=0\end{cases}}.}$

Since the value at f(0) is a removable discontinuity,

${\displaystyle \lim _{x\to a}f(x)=0}$ for all ${\displaystyle a}$.

Thus, the naïve chain rule would suggest that the limit of f(f(x)) is 0. However, it is the case that

${\displaystyle f(f(x))={\begin{cases}1&{\text{if }}x\neq 0\\0&{\text{if }}x=0\end{cases}}}$

and so

${\displaystyle \lim _{x\to a}f(f(x))=1}$ for all ${\displaystyle a}$.

For ${\displaystyle n}$ a nonnegative integer and constants ${\displaystyle a_{1},a_{2},a_{3},\ldots ,a_{n}}$ and ${\displaystyle b_{1},b_{2},b_{3},\ldots ,b_{n}}$,

• ${\displaystyle \lim _{x\to \infty }{\frac {a_{1}{x}^{n}+a_{2}{x}^{n-1}+a_{3}{x}^{n-2}+...+a_{n}}{b_{1}{x}^{n}+b_{2}{x}^{n-1}+b_{3}{x}^{n-2}+...+b_{n}}}={\frac {a_{1}}{b_{1}}}}$

This can be proven by dividing both the numerator and denominator by ${\displaystyle x^{n}}$. If the numerator is a polynomial of higher degree, the limit does not exist. If the denominator is of higher degree, the limit is 0.

• ${\displaystyle \lim _{x\to 0}{\frac {\sin x}{x}}=1}$
• ${\displaystyle \lim _{x\to 0}{\frac {1-\cos x}{x}}=0}$

• ${\displaystyle \lim _{x\to 0}(1+x)^{\frac {1}{x}}=\lim _{r\to \infty }\left(1+{\frac {1}{r}}\right)^{r}=e}$
• ${\displaystyle \lim _{x\to 0}{\frac {e^{x}-1}{x}}=1}$
• ${\displaystyle \lim _{x\to 0}{\frac {e^{ax}-1}{bx}}={\frac {a}{b}}}$
• ${\displaystyle \lim _{x\to 0}{\frac {c^{ax}-1}{bx}}={\frac {a}{b}}\ln c}$
• ${\displaystyle \lim _{x\to 0^{+}}x^{x}=1}$

• ${\displaystyle \lim _{x\to 0}{\frac {\ln(1+x)}{x}}=1}$
• ${\displaystyle \lim _{x\to 0}{\frac {\ln(1+ax)}{bx}}={\frac {a}{b}}}$
• ${\displaystyle \lim _{x\to 0}{\frac {\log _{c}(1+ax)}{bx}}={\frac {a}{b\ln c}}}$

This rule uses derivatives to find limits of indeterminate forms 0/0 or ±∞/∞, and only applies to such cases. Other indeterminate forms may be manipulated into this form. Given two functions f(x) and g(x), defined over an open interval I containing the desired limit point c, then if:

1. ${\displaystyle \lim _{x\to c}f(x)=\lim _{x\to c}g(x)=0,}$ or ${\displaystyle \lim _{x\to c}f(x)=\pm \lim _{x\to c}g(x)=\pm \infty }$, and
2. ${\displaystyle f}$ and ${\displaystyle g}$ are differentiable over ${\displaystyle I\setminus \{c\}}$, and
3. ${\displaystyle g'(x)\neq 0}$ for all ${\displaystyle x\in I\setminus \{c\}}$, and
4. ${\displaystyle \lim _{x\to c}{\frac {f'(x)}{g'(x)}}}$ exists,

then:

${\displaystyle \lim _{x\to c}{\frac {f(x)}{g(x)}}=\lim _{x\to c}{\frac {f'(x)}{g'(x)}}}$.

Normally, the first condition is the most important one.

For example: ${\displaystyle \lim _{x\to 0}{\frac {\sin(2x)}{\sin(3x)}}=\lim _{x\to 0}{\frac {2\cos(2x)}{3\cos(3x)}}={\frac {2\cdot 1}{3\cdot 1}}={\frac {2}{3}}.}$

Specifying an infinite bound on a summation or integral is a common shorthand for specifying a limit.

A short way to write the limit ${\displaystyle \lim _{n\to \infty }\sum _{i=s}^{n}f(i)}$ is ${\displaystyle \sum _{i=s}^{\infty }f(i)}$. An important example of limits of sums such as these are series.

A short way to write the limit ${\displaystyle \lim _{x\to \infty }\int _{a}^{x}f(t)\;dt}$ is ${\displaystyle \int _{a}^{\infty }f(t)\;dt}$.