Limit of a function

<math>x</math>	<math>\frac{\sin x}{x}</math>
1	0.841471...
0.1	0.998334...
0.01	0.999983...

Although the function Template:Tmath is not defined at zero, as Template:Mvar becomes closer and closer to zero, Template:Tmath becomes arbitrarily close to 1. In other words, the limit of Template:Tmath as Template:Mvar approaches zero, equals 1.

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• Fundamental theorem

• Limits
• Continuity

• Rolle's theorem
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• Derivative (generalizations)
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• Sum
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• Lists of integrals
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• Parts
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• Summand limit (term test)
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• Calculus on Euclidean space
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• Fractional
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• Precalculus
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}} 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 Template:Mvar assigns an output Template:Math to every input Template:Mvar. We say that the function has a limit Template:Mvar at an input Template:Mvar, if Template:Math gets closer and closer to Template:Mvar as Template:Mvar moves closer and closer to Template:Mvar. More specifically, the output value can be made arbitrarily close to Template:Mvar if the input to Template:Mvar is taken sufficiently close to Template:Mvar. On the other hand, if some inputs very close to Template:Mvar 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.<ref>Template:Citation</ref> 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.<ref>Template:Cite journal</ref><ref>Template:Cite journal</ref>

In his 1821 book {{#invoke:Lang|lang}}, Augustin-Louis Cauchy discussed variable quantities, infinitesimals and limits, and defined continuity of <math>y=f(x)</math> by saying that an infinitesimal change in Template:Mvar necessarily produces an infinitesimal change in Template:Mvar, while Grabiner claims that he used a rigorous epsilon-delta definition in proofs.<ref name="Grabiner1983">Template:Citation, collected in Who Gave You the Epsilon?, Template:Isbn pp. 5–13. Also available at: http://www.maa.org/pubs/Calc_articles/ma002.pdf</ref> In 1861, Karl Weierstrass first introduced the epsilon-delta definition of limit in the form it is usually written today.<ref>Template:Citation</ref> He also introduced the notations <math display="inline">\lim</math> and <math display="inline">\textstyle \lim_{x \to x_0} \displaystyle.</math><ref>Template:Citation</ref>

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.<ref>Template:Citation</ref>

Motivation

Imagine a person walking on a landscape represented by the graph Template:Math. Their horizontal position is given by Template:Mvar, much like the position given by a map of the land or by a global positioning system. Their altitude is given by the coordinate Template:Mvar. Suppose they walk towards a position Template:Math, as they get closer and closer to this point, they will notice that their altitude approaches a specific value Template:Mvar. If asked about the altitude corresponding to Template:Math, they would reply by saying Template:Math.

What, then, does it mean to say, their altitude is approaching Template:Mvar? It means that their altitude gets nearer and nearer to Template:Mvar—except for a possible small error in accuracy. For example, suppose we set a particular accuracy goal for our traveler: they must get within ten meters of Template:Mvar. They report back that indeed, they can get within ten vertical meters of Template:Mvar, arguing that as long as they are within fifty horizontal meters of Template:Mvar, their altitude is always within ten meters of Template:Mvar.

The accuracy goal is then changed: can they get within one vertical meter? Yes, supposing that they are able to move within five horizontal meters of Template:Mvar, their altitude will always remain within one meter from the target altitude Template:Mvar. Summarizing the aforementioned concept we can say that the traveler's altitude approaches Template:Mvar as their horizontal position approaches Template:Mvar, so as to say that for every target accuracy goal, however small it may be, there is some neighbourhood of Template:Mvar where all (not just some) altitudes correspond to all the horizontal positions, except maybe the horizontal position Template:Mvar itself, in that neighbourhood fulfill that accuracy goal.

The initial informal statement can now be explicated:

In fact, this explicit statement is quite close to the formal definition of the limit of a function, with values in a topological space.

More specifically, to say that

<math display=block> \lim_{x \to p}f(x) = L,</math>

is to say that Template:Math can be made as close to Template:Mvar as desired, by making Template:Mvar close enough, but not equal, to Template:Mvar.

The following definitions, known as Template:Math-definitions, are the generally accepted definitions for the limit of a function in various contexts.

Functions of a single variable

Template:Math-definition of limit

File:Epsilon-delta limit.svg

For the depicted Template:Mvar, Template:Mvar, and Template:Mvar, we can ensure that the value Template:Math is within an arbitrarily small interval Template:Math by restricting Template:Mvar to a sufficiently small interval Template:Math Hence Template:Math as Template:Math.

Suppose <math>f: \R \rightarrow \R</math> is a function defined on the real line, and there are two real numbers Template:Mvar and Template:Mvar. One would say: The limit of Template:Mvar of Template:Mvar, as Template:Mvar approaches Template:Mvar, exists, and it equals Template:Mvar. and write,<ref name="swokowski">Template:Citation</ref>

<math display=block> \lim_{x \to p} f(x) = L,</math>

or alternatively, say Template:Math tends to Template:Mvar as Template:Mvar tends to Template:Mvar, and write,

<math display=block> f(x) \to L \text{ as } x \to p,</math>

if the following property holds: for every real Template:Math, there exists a real Template:Math such that for all real Template:Mvar, Template:Math implies Template:Math.<ref name="swokowski" /> Symbolically, <math display=block>(\forall \varepsilon > 0 ) \, (\exists \delta > 0) \, (\forall x \in \R) \, (0 < |x - p| < \delta \implies |f(x) - L| < \varepsilon).</math>

For example, we may say <math display=block>\lim_{x \to 2} (4x + 1) = 9</math> because for every real Template:Math, we can take Template:Math, so that for all real Template:Mvar, if Template:Math, then Template:Math.

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

Or, symbolically: <math display=block>(\forall \varepsilon > 0 ) \, (\exists \delta > 0) \, (\forall x \in (a, b)) \, (0 < |x - p| < \delta \implies |f(x) - L| < \varepsilon).</math>

For example, we may say <math display=block>\lim_{x \to 1} \sqrt{x+3} = 2</math> because for every real Template:Math, we can take Template:Math, so that for all real Template:Math, if Template:Math, then Template:Math. In this example, Template:Math 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 Template:Mvar being defined at Template:Mvar, nor on the value Template:Math—if it is defined. For example, let <math>f: [0,1)\cup (1,2] \to \R, f(x) = \tfrac{2x^2 - x - 1}{x-1}.</math> <math display=block>\lim_{x \to 1} f(x) = 3</math> because for every Template:Math, we can take Template:Math, so that for all real Template:Math, if Template:Math, then Template:Math. Note that here Template:Math is undefined.

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

The letters Template:Mvar and Template:Mvar can be understood as "error" and "distance". In fact, Cauchy used Template:Mvar as an abbreviation for "error" in some of his work,<ref name="Grabiner1983" /> though in his definition of continuity, he used an infinitesimal <math>\alpha</math> rather than either Template:Mvar or Template:Mvar (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 Template:Mvar and Template:Mvar represent distances helps suggest these generalizations.

Existence and one-sided limits

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}

File:Upper semi.svg

The limit as <math>x \to x_0^+</math> differs from that as <math>x \to x_0^-.</math> Therefore, the limit as Template:Math does not exist.

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

<math display=block> \lim_{x \to p^+}f(x) = L </math>

or

<math display=block> \lim_{x \to p^-}f(x) = L </math>

File:Undefined limit examples.png

The first three functions have points for which the limit does not exist, while the function<math display="block"> f(x) = \frac{\sin(x)}{x} </math>is not defined at <math>x = 0</math>, but its limit does exist.

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

A formal definition is as follows. The limit of Template:Mvar as Template:Mvar approaches Template:Mvar from above is Template:Mvar if:

For every Template:Math, there exists a Template:Math such that whenever Template:Math, we have Template:Math.

<math display=block>(\forall \varepsilon > 0 ) \, (\exists \delta > 0) \, (\forall x \in (a,b))\, (0 < x - p < \delta \implies |f(x) - L| < \varepsilon).</math>

The limit of Template:Mvar as Template:Mvar approaches Template:Mvar from below is Template:Mvar if:

For every Template:Math, there exists a Template:Math such that whenever Template:Math, we have Template:Math.

<math display=block>(\forall \varepsilon > 0 )\, (\exists \delta > 0) \, (\forall x \in (a,b)) \, (0 < p - x < \delta \implies |f(x) - L| < \varepsilon).</math>

If the limit does not exist, then the oscillation of Template:Mvar at Template:Mvar is non-zero.

More general definition using limit points and subsets

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

In general:<ref>Template:Harv</ref> Let <math>f : S \to \R</math> be a real-valued function defined on some <math>S \subseteq \R.</math> Let Template:Mvar be a limit point of some <math>T \subset S</math>—that is, Template:Mvar is the limit of some sequence of elements of Template:Mvar distinct from Template:Mvar. Then we say the limit of Template:Mvar, as Template:Mvar approaches Template:Mvar from values in Template:Mvar, is Template:Mvar, written <math display=block>\lim_{ {x \to p} \atop {x \in T} } f(x) = L</math> if the following holds: Template:Block indent <math display=block>(\forall \varepsilon > 0 )\, (\exists \delta > 0) \,(\forall x \in T)\, (0 < |x - p| < \delta \implies |f(x) - L| < \varepsilon).</math>

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

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

Notably, the previous two-sided definition works on <math>\operatorname{int} S \cup \operatorname{iso} S^c,</math> which is a subset of the limit points of Template:Mvar.

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

Deleted versus non-deleted limits

The definition of limit given here does not depend on how (or whether) Template:Mvar is defined at Template:Mvar. Bartle<ref name="Bartle 1967">Template:Harvtxt</ref> refers to this as a deleted limit, because it excludes the value of Template:Mvar at Template:Mvar. The corresponding non-deleted limit does depend on the value of Template:Mvar at Template:Mvar, if Template:Mvar is in the domain of Template:Mvar. Let <math>f : S \to \R</math> be a real-valued function. The non-deleted limit of Template:Mvar, as Template:Mvar approaches Template:Mvar, is Template:Mvar if

Template:Block indent <math display=block>(\forall \varepsilon > 0 )\, (\exists \delta > 0) \, (\forall x \in S)\, (|x - p| < \delta \implies |f(x) - L| < \varepsilon).</math>

The definition is the same, except that the neighborhood Template:Math now includes the point Template:Mvar, in contrast to the deleted neighborhood Template:Math. 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).<ref>Template:Harvtxt</ref>

Bartle<ref name="Bartle 1967"/> notes that although by "limit" some authors do mean this non-deleted limit, deleted limits are the most popular.<ref>For example, Template:Harvtxt, Template:Harvtxt, Template:Harvtxt, Template:Harvtxt, Template:Harvtxt all take "limit" to mean the deleted limit.</ref>

Examples

Non-existence of one-sided limit(s)

File:Discontinuity essential.svg

Function without a limit at an essential discontinuity

The function <math display=block>f(x)=\begin{cases} \sin\frac{5}{x-1} & \text{ for } x<1 \\ 0 & \text{ for } x=1 \\[2pt] \frac{1}{10x-10}& \text{ for } x>1 \end{cases}</math> has no limit at Template:Math (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, see picture), but has a limit at every other Template:Mvar-coordinate.

The function <math display=block>f(x)=\begin{cases} 1 & x \text{ rational } \\ 0 & x \text{ irrational } \end{cases}</math> (a.k.a., the Dirichlet function) has no limit at any Template:Mvar-coordinate.

Non-equality of one-sided limits

The function <math display=block>f(x)=\begin{cases} 1 & \text{ for } x < 0 \\ 2 & \text{ for } x \ge 0 \end{cases}</math> has a limit at every non-zero Template:Mvar-coordinate (the limit equals 1 for negative Template:Mvar and equals 2 for positive Template:Mvar). The limit at Template:Math does not exist (the left-hand limit equals 1, whereas the right-hand limit equals 2).

Limits at only one point

The functions <math display=block>f(x)=\begin{cases} x & x \text{ rational } \\ 0 & x \text{ irrational } \end{cases}</math> and <math display=block>f(x)=\begin{cases} |x| & x \text{ rational } \\ 0 & x \text{ irrational } \end{cases}</math> both have a limit at Template:Math and it equals 0.

Limits at countably many points

The function <math display=block>f(x)=\begin{cases} \sin x & x \text{ irrational } \\

1     &  x \text{ rational }

\end{cases}</math> has a limit at any Template:Mvar-coordinate of the form <math>\tfrac{\pi}{2} + 2n\pi,</math> where Template:Mvar is any integer.

Limits involving infinity

Limits at infinity

File:Limit Infinity SVG.svg

The limit of this function at infinity exists

Let <math>f:S \to \R</math> be a function defined on <math>S \subseteq \R.</math> The limit of Template:Mvar as Template:Mvar approaches infinity is Template:Mvar, denoted

<math display=block> \lim_{x \to \infty}f(x) = L,</math>

means that: Template:Block indent <math display=block>(\forall \varepsilon > 0 )\, (\exists c > 0) \,(\forall x \in S) \,(x > c \implies |f(x) - L| < \varepsilon).</math>

Similarly, the limit of Template:Mvar as Template:Mvar approaches minus infinity is Template:Mvar, denoted

<math display=block> \lim_{x \to -\infty}f(x) = L,</math>

means that: Template:Block indent <math display=block>(\forall \varepsilon > 0)\, (\exists c > 0) \,(\forall x \in S)\, (x < -c \implies |f(x) - L| < \varepsilon).</math>

For example, <math display=block> \lim_{x \to \infty} \left(-\frac{3\sin x}{x} + 4\right) = 4</math> because for every Template:Math, we can take Template:Math such that for all real Template:Mvar, if Template:Math, then Template:Math.

Another example is that <math display=block> \lim_{x \to -\infty}e^{x} = 0</math> because for every Template:Math, we can take Template:Math such that for all real Template:Mvar, if Template:Math, then Template:Math.

Infinite limits

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 <math>f:S \to\mathbb{R}</math> be a function defined on <math>S\subseteq\mathbb{R}.</math> The statement the limit of Template:Mvar as Template:Mvar approaches Template:Mvar is infinity, denoted

<math display=block> \lim_{x \to p} f(x) = \infty, </math>

means that: Template:Block indent <math display=block>(\forall N > 0)\, (\exists \delta > 0)\, (\forall x \in S)\, (0 < | x-p | < \delta \implies f(x) > N) .</math>

The statement the limit of Template:Mvar as Template:Mvar approaches Template:Mvar is minus infinity, denoted

<math display=block> \lim_{x \to p} f(x) = -\infty, </math>

means that: Template:Block indent <math display=block>(\forall N > 0) \, (\exists \delta > 0) \, (\forall x \in S)\, (0 < | x-p | < \delta \implies f(x) < -N) .</math>

For example, <math display=block>\lim_{x \to 1} \frac{1}{(x-1)^2} = \infty</math> because for every Template:Math, we can take <math display="inline">\delta = \tfrac{1}{\sqrt{N}\delta} = \tfrac{1}{\sqrt N}</math> such that for all real Template:Math, if Template:Math, then Template:Math.

These ideas can be used together to produce definitions for different combinations, such as

<math display=block> \lim_{x \to \infty} f(x) = \infty,</math> or <math> \lim_{x \to p^+}f(x) = -\infty.</math>

For example, <math display=block>\lim_{x \to 0^+} \ln x = -\infty</math> because for every Template:Math, we can take Template:Math such that for all real Template:Math, if Template:Math, then Template:Math.

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 Template:Math for some Template:Tmath
• a neighborhood of ∞ is defined to contain an interval Template:Math where Template:Tmath and
• a neighborhood of Template:Tmath is defined in the normal way metric space Template:Tmath

In this case, Template:Tmath is a topological space and any function of the form <math>f : X \to Y</math> with <math>X, Y \subseteq \overline \R</math> 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.

Alternative notation

Many authors<ref>For example, Limit at Encyclopedia of Mathematics</ref> 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 Template:Tmath and the projectively extended real line is Template:Tmath where a neighborhood of ∞ is a set of the form <math>\{x: |x| > c\}.</math> 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, <math>x^{-1}</math> does not possess a central limit (which is normal):

<math display=block>\lim_{x \to 0^{+}}{1\over x} = +\infty, \quad \lim_{x \to 0^{-}}{1\over x} = -\infty.</math>

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

<math display=block>\lim_{x \to 0^{+}}{1\over x} = \lim_{x \to 0^{-}}{1\over x} = \lim_{x \to 0}{1\over x} = \infty.</math>

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 <math>\lim_{x \to 0^{-}}{x^{-1}} = -\infty,</math> namely, it is convenient for <math>\lim_{x \to -\infty}{x^{-1}} = -0</math> to be considered true. Such zeroes can be seen as an approximation to infinitesimals.

Limits at infinity for rational functions

File:Tamasol SVG.svg

Horizontal asymptote about Template:Math

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

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

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

Functions of more than one variable

Ordinary limits

By noting that Template:Math represents a distance, the definition of a limit can be extended to functions of more than one variable. In the case of a function <math>f : S \times T \to \R</math> defined on <math>S \times T \subseteq \R^2,</math> we defined the limit as follows: the limit of Template:Mvar as Template:Math approaches Template:Math is Template:Mvar, written

<math display=block> \lim_{(x,y) \to (p, q)} f(x, y) = L </math>

if the following condition holds:

For every Template:Math, there exists a Template:Math such that for all Template:Mvar in Template:Mvar and Template:Mvar in Template:Mvar, whenever <math display=inline>0 < \sqrt{(x-p)^2 + (y-q)^2} < \delta,</math> we have Template:Math,<ref>Template:Citation</ref>

or formally: <math display=block>(\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)).</math>

Here <math display=inline>\sqrt{(x-p)^2 + (y-q)^2}</math> is the Euclidean distance between Template:Math and Template:Math. (This can in fact be replaced by any norm Template:Math, and be extended to any number of variables.)

For example, we may say <math display=block> \lim_{(x,y) \to (0, 0)} \frac{x^4}{x^2+y^2} = 0 </math> because for every Template:Math, we can take <math display=inline>\delta = \sqrt \varepsilon</math> such that for all real Template:Math and real Template:Math, if <math display=inline>0 < \sqrt{(x-0)^2 + (y-0)^2} < \delta,</math> then Template:Math.

Similar to the case in single variable, the value of Template:Mvar at Template:Math does not matter in this definition of limit.

For such a multivariable limit to exist, this definition requires the value of Template:Mvar approaches Template:Mvar along every possible path approaching Template:Math.Template:Sfnp In the above example, the function <math display=block>f(x, y) = \frac{x^4}{x^2+y^2}</math> satisfies this condition. This can be seen by considering the polar coordinates <math display=block>(x,y) = (r\cos\theta, r\sin\theta) \to (0, 0),</math> which gives <math display=block>\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.</math> Here Template:Math is a function of r which controls the shape of the path along which Template:Mvar is approaching Template:Math. Since Template:Math is bounded between [−1, 1], by the sandwich theorem, this limit tends to 0.

In contrast, the function <math display=block>f(x, y) = \frac{xy}{x^2 + y^2}</math> does not have a limit at Template:Math. Taking the path Template:Math, we obtain <math display=block>\lim_{t \to 0} f(t, 0) = \lim_{t \to 0} \frac{0}{t^2} = 0,</math> while taking the path Template:Math, we obtain <math display=block>\lim_{t \to 0} f(t, t) = \lim_{t \to 0} \frac{t^2}{t^2 + t^2} = \frac{1}{2}.</math>

Since the two values do not agree, Template:Mvar does not tend to a single value as Template:Math approaches Template:Math.

Multiple limits

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.<ref name="Zakon_219">Template:Citation</ref> Let <math>f : S \times T \to \R</math> be defined on <math>S \times T \subseteq \R^2,</math> we say the double limit of Template:Mvar as Template:Mvar approaches Template:Mvar and Template:Mvar approaches Template:Mvar is Template:Mvar, written

<math display=block> \lim_{ {x \to p} \atop {y \to q} } f(x, y) = L </math>

if the following condition holds: Template:Block indent <math display=block>(\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) .</math>

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

If the domain of Template:Mvar is restricted to <math>(S\setminus\{p\}) \times (T\setminus\{q\}),</math> then the two definitions of limits coincide.<ref name="Zakon_219" />

Multiple limits at infinity

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

if the following condition holds: Template:Block indent <math display=block>(\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) .</math>

We say the double limit of Template:Mvar as Template:Mvar and Template:Mvar approaches minus infinity is Template:Mvar, written <math display=block> \lim_{ {x \to -\infty} \atop {y \to -\infty} } f(x, y) = L </math>

if the following condition holds: Template:Block indent <math display=block>(\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) .</math>

Pointwise limits and uniform limits

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Let <math>f : S \times T \to \R.</math> Instead of taking limit as Template:Math, we may consider taking the limit of just one variable, say, Template:Math, to obtain a single-variable function of Template:Mvar, namely <math>g : T \to \R.</math> 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 Template:Mvar as Template:Mvar approaches Template:Mvar is Template:Mvar, denoted <math display=block>\lim_{x\to p}f(x, y) = g(y),</math> or <math display=block>\lim_{x \to p}f(x, y) = g(y) \;\; \text{pointwise}.</math>

Alternatively, we may say Template:Mvar tends to Template:Mvar pointwise as Template:Mvar approaches Template:Mvar, denoted <math display=block>f(x, y) \to g(y) \;\; \text{as} \;\; x \to p,</math> or <math display=block>f(x, y) \to g(y) \;\; \text{pointwise} \;\; \text{as} \;\; x \to p.</math>

This limit exists if the following holds: Template:Block indent <math display=block>(\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) .</math>

Here, Template:Math is a function of both Template:Mvar and Template:Mvar. Each Template:Mvar is chosen for a specific point of Template:Mvar. Hence we say the limit is pointwise in Template:Mvar. For example, <math display=block>f(x, y) = \frac{x}{\cos y}</math> has a pointwise limit of constant zero function <math display=block>\lim_{x \to 0}f(x, y) = 0(y) \;\; \text{pointwise}</math> because for every fixed Template:Mvar, the limit is clearly 0. This argument fails if Template:Mvar is not fixed: if Template:Mvar is very close to Template:Math, 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 Template:Mvar on Template:Mvar as Template:Mvar approaches Template:Mvar is Template:Mvar, denoted <math display=block>\underset{ {x \to p} \atop {y \in T} }{\mathrm{unif} \lim \;} f(x, y) = g(y),</math> or <math display=block>\lim_{x \to p}f(x, y) = g(y) \;\; \text{uniformly on} \; T.</math>

Alternatively, we may say Template:Mvar tends to Template:Mvar uniformly on Template:Mvar as Template:Mvar approaches Template:Mvar, denoted <math display=block>f(x, y) \rightrightarrows g(y) \; \text{on} \; T \;\; \text{as} \;\; x \to p,</math> or <math display=block>f(x, y) \to g(y) \;\; \text{uniformly on}\; T \;\; \text{as} \;\; x \to p.</math>

This limit exists if the following holds: Template:Block indent <math display=block>(\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) .</math>

Here, Template:Math is a function of only Template:Mvar but not Template:Mvar. In other words, δ is uniformly applicable to all Template:Mvar in Template:Mvar. Hence we say the limit is uniform in Template:Mvar. For example, <math display=block>f(x, y) = x \cos y</math> has a uniform limit of constant zero function <math display=block>\lim_{x \to 0}f(x, y) = 0(y) \;\; \text{ uniformly on}\; \R</math> because for all real Template:Mvar, Template:Math is bounded between Template:Math. Hence no matter how Template:Mvar behaves, we may use the sandwich theorem to show that the limit is 0.

Iterated limits

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Let <math>f : S \times T \to \R.</math> We may consider taking the limit of just one variable, say, Template:Math, to obtain a single-variable function of Template:Mvar, namely <math>g : T \to \R,</math> and then take limit in the other variable, namely Template:Math, to get a number Template:Mvar. Symbolically, <math display=block>\lim_{y \to q} \lim_{x \to p} f(x, y) = \lim_{y \to q} g(y) = L.</math>

This limit is known as iterated limit of the multivariable function.Template:Sfnp The order of taking limits may affect the result, i.e.,

<math display=block>\lim_{y \to q} \lim_{x \to p} f(x,y) \ne \lim_{x \to p} \lim_{y \to q} f(x, y)</math> in general.

A sufficient condition of equality is given by the Moore-Osgood theorem, which requires the limit <math>\lim_{x \to p}f(x, y) = g(y)</math> to be uniform on Template:Mvar.<ref>Template:Citation</ref>

Functions on metric spaces

Suppose Template:Mvar and Template:Mvar are subsets of metric spaces Template:Mvar and Template:Mvar, respectively, and Template:Math is defined between Template:Mvar and Template:Mvar, with Template:Math, Template:Mvar a limit point of Template:Mvar and Template:Math. It is said that the limit of Template:Mvar as Template:Mvar approaches Template:Mvar is Template:Mvar and write

<math display=block> \lim_{x \to p}f(x) = L </math>

if the following property holds:

Template:Block indent <math display=block>(\forall \varepsilon > 0 )\, (\exists \delta > 0) \,(\forall x \in M) \,(0 < d_A(x, p) < \delta \implies d_B(f(x), L) < \varepsilon).</math>

Again, note that Template:Mvar need not be in the domain of Template:Mvar, nor does Template:Mvar need to be in the range of Template:Mvar, and even if Template:Math is defined it need not be equal to Template:Mvar.

Euclidean metric

The limit in Euclidean space is a direct generalization of limits to vector-valued functions. For example, we may consider a function <math>f:S \times T \to \R^3</math> such that <math display=block>f(x, y) = (f_1(x, y), f_2(x, y), f_3(x, y) ).</math> Then, under the usual Euclidean metric, <math display=block>\lim_{(x, y) \to (p, q)} f(x, y) = (L_1, L_2, L_3)</math> if the following holds:

Template:Block indent <math display=block>(\forall \varepsilon > 0 )\, (\exists \delta > 0) \, (\forall x \in S) \, (\forall y \in T)\, \left(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 \right).</math>

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:<ref name="Hartman" /> <math display=block>\lim_{(x, y) \to (p, q)} \Bigl(f_1(x, y), f_2(x, y), f_3(x, y)\Bigr) = \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).</math>

Manhattan metric

One might also want to consider spaces other than Euclidean space. An example would be the Manhattan space. Consider <math>f:S \to \R^2</math> such that <math display=block>f(x) = (f_1(x), f_2(x)).</math> Then, under the Manhattan metric, <math display=block>\lim_{x \to p} f(x) = (L_1, L_2)</math> if the following holds:

Template:Block indent <math display=block>(\forall \varepsilon > 0 )\, (\exists \delta > 0) \,(\forall x \in S) \,(0 < |x - p| < \delta \implies |f_1 - L_1| + |f_2 - L_2| < \varepsilon).</math>

Since this is also a finite-dimension vector-valued function, the limit theorem stated above also applies.Template:Sfnp

Uniform metric

Finally, we will discuss the limit in function space, which has infinite dimensions. Consider a function Template:Math in the function space <math>S \times T \to \R.</math> We want to find out as Template:Mvar approaches Template:Mvar, how Template:Math will tend to another function Template:Math, which is in the function space <math>T \to \R.</math> The "closeness" in this function space may be measured under the uniform metric.<ref>Template:Citation</ref> Then, we will say the uniform limit of Template:Mvar on Template:Mvar as Template:Mvar approaches Template:Mvar is Template:Mvar and write <math display=block>\underset{ {x \to p} \atop {y \in T} }{\mathrm{unif} \lim \;} f(x, y) = g(y),</math> or <math display=block>\lim_{x \to p}f(x, y) = g(y) \;\; \text{uniformly on} \; T,</math>

if the following holds:

Template:Block indent <math display=block>(\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).</math>

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.

Functions on topological spaces

Suppose Template:Mvar and Template:Mvar are topological spaces with Template:Mvar a Hausdorff space. Let Template:Mvar be a limit point of Template:Math, and Template:Math. For a function Template:Math, it is said that the limit of Template:Mvar as Template:Mvar approaches Template:Mvar is Template:Mvar, written

<math display=block> \lim_{x \to p}f(x) = L ,</math>

if the following property holds:

This last part of the definition can also be phrased "there exists an open punctured neighbourhood Template:Mvar of Template:Mvar such that Template:Math".

The domain of Template:Mvar does not need to contain Template:Mvar. If it does, then the value of Template:Mvar at Template:Mvar is irrelevant to the definition of the limit. In particular, if the domain of Template:Mvar is Template:Math (or all of Template:Mvar), then the limit of Template:Mvar as Template:Math exists and is equal to Template:Mvar if, for all subsets Template:Math of Template:Mvar with limit point Template:Mvar, the limit of the restriction of Template:Mvar to Template:Math exists and is equal to Template:Mvar. Sometimes this criterion is used to establish the non-existence of the two-sided limit of a function on Template:Tmath by showing that the one-sided limits either fail to exist or do not agree. Such a view is fundamental in the field of general topology, where limits and continuity at a point are defined in terms of special families of subsets, called filters, or generalized sequences known as nets.

Alternatively, the requirement that Template:Mvar be a Hausdorff space can be relaxed to the assumption that Template:Mvar be a general topological space, but then the limit of a function may not be unique. In particular, one can no longer talk about the limit of a function at a point, but rather a limit or the set of limits at a point.

A function is continuous at a limit point Template:Mvar of and in its domain if and only if Template:Math is the (or, in the general case, a) limit of Template:Math as Template:Mvar tends to Template:Mvar.

There is another type of limit of a function, namely the sequential limit. Let Template:Math be a mapping from a topological space Template:Mvar into a Hausdorff space Template:Mvar, Template:Math a limit point of Template:Mvar and Template:Math. The sequential limit of Template:Mvar as Template:Mvar tends to Template:Mvar is Template:Mvar if Template:Anchor

For every sequence (Template:Mvar) in Template:Math that converges to Template:Mvar, the sequence Template:Math converges to Template:Mvar.

If Template:Mvar is the limit (in the sense above) of Template:Mvar as Template:Mvar approaches Template:Mvar, then it is a sequential limit as well, however the converse need not hold in general. If in addition Template:Mvar is metrizable, then Template:Mvar is the sequential limit of Template:Mvar as Template:Mvar approaches Template:Mvar if and only if it is the limit (in the sense above) of Template:Mvar as Template:Mvar approaches Template:Mvar.

Other characterizations

In terms of sequences

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: <math display=block>\lim_{x\to a}f(x)=L</math> if, and only if, for all sequences Template:Mvar (with, for all Template:Mvar, Template:Mvar not equal to Template:Mvar) converging to Template:Mvar the sequence Template:Math converges to Template:Mvar. 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 Template:Mvar to converge to Template:Mvar 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 Template:Mvar be a real-valued function with the domain Template:Math. Let Template:Mvar be the limit of a sequence of elements of Template:Math Then the limit (in this sense) of Template:Mvar is Template:Mvar as Template:Mvar approaches Template:Mvar if for every sequence Template:Math (so that for all Template:Mvar, Template:Mvar is not equal to Template:Mvar) that converges to Template:Mvar, the sequence Template:Math converges to Template:Mvar. This is the same as the definition of a sequential limit in the preceding section obtained by regarding the subset Template:Math of Template:Tmath as a metric space with the induced metric.

In non-standard calculus

In non-standard calculus the limit of a function is defined by: <math display=block>\lim_{x\to a}f(x)=L</math> if and only if for all <math>x\in \R^*,</math> <math>f^*(x)-L</math> is infinitesimal whenever Template:Math is infinitesimal. Here <math>\R^*</math> are the hyperreal numbers and Template:Mvar is the natural extension of Template:Mvar to the non-standard real numbers. Keisler proved that such a hyperreal definition of limit reduces the quantifier complexity by two quantifiers.<ref>Template:Citation</ref> 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.<ref>Template:Citation</ref> 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".<ref>Template:Citation</ref>

In terms of nearness

At the 1908 international congress of mathematics F. Riesz introduced an alternate way defining limits and continuity in concept called "nearness".<ref>Template:Citation</ref> A point Template:Mvar is defined to be near a set <math>A\subseteq \R</math> if for every Template:Math there is a point Template:Math so that Template:Math. In this setting the <math display=block>\lim_{x\to a} f(x)=L</math> if and only if for all <math>A\subseteq \R,</math> Template:Mvar is near Template:Math whenever Template:Mvar is near Template:Mvar. Here Template:Math is the set <math>\{f(x) | x \in A\}.</math> This definition can also be extended to metric and topological spaces.

Relationship to continuity

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The notion of the limit of a function is very closely related to the concept of continuity. A function Template:Mvar is said to be continuous at Template:Mvar if it is both defined at Template:Mvar and its value at Template:Mvar equals the limit of Template:Mvar as Template:Mvar approaches Template:Mvar:

<math display="block">\lim_{x\to c} f(x) = f(c).</math> We have here assumed that Template:Mvar is a limit point of the domain of Template:Mvar.

Properties

If a function Template:Mvar is real-valued, then the limit of Template:Mvar at Template:Mvar is Template:Mvar if and only if both the right-handed limit and left-handed limit of Template:Mvar at Template:Mvar exist and are equal to Template:Mvar.Template:Sfnp

The function Template:Mvar is continuous at Template:Mvar if and only if the limit of Template:Math as Template:Mvar approaches Template:Mvar exists and is equal to Template:Math. If Template:Math is a function between metric spaces Template:Mvar and Template:Mvar, then it is equivalent that Template:Mvar transforms every sequence in Template:Mvar which converges towards Template:Mvar into a sequence in Template:Mvar which converges towards Template:Math.

If Template:Mvar is a normed vector space, then the limit operation is linear in the following sense: if the limit of Template:Math as Template:Mvar approaches Template:Mvar is Template:Mvar and the limit of Template:Math as Template:Mvar approaches Template:Mvar is Template:Mvar, then the limit of Template:Math as Template:Mvar approaches Template:Mvar is Template:Math. If Template:Mvar is a scalar from the base field, then the limit of Template:Math as Template:Mvar approaches Template:Mvar is Template:Mvar.

If Template:Mvar and Template:Mvar are real-valued (or complex-valued) functions, then taking the limit of an operation on Template:Math and Template:Math (e.g., Template:Math, Template:Math, Template:Math, Template:Math, Template:Mvar) under certain conditions is compatible with the operation of limits of Template:Math and Template:Math. This fact is often called the algebraic limit theorem. The main condition needed to apply the following rules is that the limits on the right-hand sides of the equations exist (in other words, these limits are finite values including 0). Additionally, the identity for division requires that the denominator on the right-hand side is non-zero (division by 0 is not defined), and the identity for exponentiation requires that the base is positive, or zero while the exponent is positive (finite).

<math display=block>\begin{array}{lcl} \displaystyle \lim_{x \to p} (f(x) + g(x)) & = & \displaystyle \lim_{x \to p} f(x) + \lim_{x \to p} g(x) \\ \displaystyle \lim_{x \to p} (f(x) - g(x)) & = & \displaystyle \lim_{x \to p} f(x) - \lim_{x \to p} g(x) \\ \displaystyle \lim_{x \to p} (f(x)\cdot g(x)) & = & \displaystyle \lim_{x \to p} f(x) \cdot \lim_{x \to p} g(x) \\ \displaystyle \lim_{x \to p} (f(x)/g(x)) & = & \displaystyle {\lim_{x \to p} f(x) / \lim_{x \to p} g(x)} \\ \displaystyle \lim_{x \to p} f(x)^{g(x)} & = & \displaystyle {\lim_{x \to p} f(x) ^ {\lim_{x \to p} g(x)}} \end{array}</math>

These rules are also valid for one-sided limits, including when Template:Mvar is ∞ or −∞. In each rule above, when one of the limits on the right is ∞ or −∞, the limit on the left may sometimes still be determined by the following rules.

<math display=block>\begin{array}{rcl}

 q + \infty & = & \infty \text{ if } q \neq -\infty \\[8pt]
 q \times \infty & = & \begin{cases} 
   \infty & \text{if } q > 0 \\ 
   -\infty & \text{if } q < 0 
   \end{cases} \\[6pt]
 \displaystyle \frac q \infty & = & 0 \text{ if } q \neq \infty \text{ and } q \neq -\infty \\[6pt]
 \infty^q & = & \begin{cases} 
   0 & \text{if } q < 0 \\
   \infty & \text{if } q > 0 
   \end{cases} \\[4pt]
 q^\infty & = & \begin{cases}
   0 & \text{if } 0 < q < 1 \\
   \infty & \text{if }  q > 1 
   \end{cases} \\[4pt]
 q^{-\infty} & = & \begin{cases} 
   \infty & \text{if } 0 < q < 1 \\
   0 & \text{if } q > 1
   \end{cases}

\end{array}</math>

(see also Extended real number line).

In other cases the limit on the left may still exist, although the right-hand side, called an indeterminate form, does not allow one to determine the result. This depends on the functions Template:Mvar and Template:Mvar. These indeterminate forms are:

<math display=block>\begin{array}{cc} \displaystyle \frac{0}{0} & \displaystyle \frac{\pm \infty}{\pm \infty} \\[6pt] 0 \times \pm \infty & \infty + -\infty \\[8pt] \qquad 0^0 \qquad & \qquad \infty^0 \qquad \\[8pt] 1^{\pm \infty} \end{array}</math>

See further L'Hôpital's rule below and Indeterminate form.

Limits of compositions of functions

In general, from knowing that <math>\lim_{y \to b} f(y) = c</math> and <math>\lim_{x \to a} g(x) = b,</math> it does not follow that <math>\lim_{x \to a} f(g(x)) = c.</math> However, this "chain rule" does hold if one of the following additional conditions holds:

• Template:Math (that is, Template:Mvar is continuous at Template:Mvar), or
• Template:Mvar does not take the value Template:Mvar near Template:Mvar (that is, there exists a Template:Math such that if Template:Math then Template:Math).

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

<math display=block>f(x) = g(x) = \begin{cases}

 0 & \text{if } x\neq 0 \\ 
 1 & \text{if } x=0 

\end{cases}</math>

Since the value at Template:Math is a removable discontinuity, <math display=block>\lim_{x \to a} f(x) = 0</math> for all Template:Mvar. Thus, the naïve chain rule would suggest that the limit of Template:Math is 0. However, it is the case that <math display=block>f(f(x))=\begin{cases}

 1 & \text{if } x\neq 0 \\ 
 0 & \text{if } x = 0 

\end{cases}</math> and so <math display=block>\lim_{x \to a} f(f(x)) = 1</math> for all Template:Mvar.

Limits of special interest

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Rational functions

For Template:Mvar a nonnegative integer and constants <math>a_1, a_2, a_3,\ldots, a_n</math> and <math>b_1, b_2, b_3,\ldots, b_n,</math>

<math display=block>\lim_{x \to \infty} \frac{a_1 x^n + a_2 x^{n-1} + a_3 x^{n-2} + \dots + a_n}{b_1 x^n + b_2 x^{n-1} + b_3 x^{n-2} + \dots + b_n} = \frac{a_1}{b_1}</math>

This can be proven by dividing both the numerator and denominator by Template:Mvar. 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.

Trigonometric functions

<math display=block>\begin{array}{lcl} \displaystyle \lim_{x \to 0} \frac{\sin x}{x} & = & 1 \\[4pt] \displaystyle \lim_{x \to 0} \frac{1 - \cos x}{x} & = & 0 \end{array}</math>

Exponential functions

<math display=block>\begin{array}{lcl}

 \displaystyle \lim_{x \to 0} (1+x)^{\frac{1}{x}} & = & \displaystyle \lim_{r \to \infty} \left(1+\frac{1}{r}\right)^r = e \\[4pt]
 \displaystyle \lim_{x \to 0} \frac{e^{x}-1}{x} & = & 1 \\[4pt]
 \displaystyle \lim_{x \to 0} \frac{e^{ax}-1}{bx} & = & \displaystyle \frac{a}{b} \\[4pt]
 \displaystyle \lim_{x \to 0} \frac{c^{ax}-1}{bx} & = & \displaystyle \frac{a}{b}\ln c \\[4pt]
 \displaystyle \lim_{x \to 0^+} x^x & = & 1

\end{array}</math>

Logarithmic functions

<math display=block>\begin{array}{lcl}

 \displaystyle \lim_{x \to 0} \frac{\ln(1+x)}{x} & = & 1 \\[4pt]
 \displaystyle \lim_{x \to 0} \frac{\ln(1+ax)}{bx} & = & \displaystyle \frac{a}{b} \\[4pt]
 \displaystyle \lim_{x \to 0} \frac{\log_c(1+ax)}{bx} & = & \displaystyle \frac{a}{b\ln c}

\end{array}</math>

L'Hôpital's rule

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} This rule uses derivatives to find limits of indeterminate forms Template:Math or Template:Math, and only applies to such cases. Other indeterminate forms may be manipulated into this form. Given two functions Template:Math and Template:Math, defined over an open interval Template:Mvar containing the desired limit point Template:Mvar, then if:

1. <math>\lim_{x \to c}f(x)=\lim_{x \to c}g(x)=0,</math> or <math>\lim_{x \to c}f(x)=\pm\lim_{x \to c}g(x) = \pm\infty,</math> and
2. <math>f</math> and <math>g</math> are differentiable over <math>I \setminus \{c\},</math> and
3. <math>g'(x)\neq 0</math> for all <math> x \in I \setminus \{c\},</math> and
4. <math>\lim_{x\to c}\tfrac{f'(x)}{g'(x)}</math> exists,

then: <math display=block>\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}.</math>

Normally, the first condition is the most important one.

For example: <math>\lim_{x \to 0} \frac{\sin (2x)}{\sin (3x)} = \lim_{x \to 0} \frac{2 \cos (2x)}{3 \cos (3x)} = \frac{2 \sdot 1}{3 \sdot 1} = \frac{2}{3}.</math>

Summations and integrals

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

A short way to write the limit <math>\lim_{n \to \infty} \sum_{i=s}^n f(i) </math> is <math>\sum_{i=s}^\infty f(i).</math> An important example of limits of sums such as these are series.

A short way to write the limit <math>\lim_{x \to \infty} \int_a^x f(t) \; dt </math> is <math>\int_a^\infty f(t) \; dt.</math>

A short way to write the limit <math>\lim_{x \to -\infty} \int_x^b f(t) \; dt </math> is <math>\int_{-\infty}^b f(t) \; dt.</math>

See also

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References

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External links

• MacTutor History of Weierstrass.
• MacTutor History of Bolzano
• Visual Calculus by Lawrence S. Husch, University of Tennessee (2001)