Limit 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 , Augustin-Louis Cauchy discussed variable quantities, infinitesimals and limits, and defined continuity of y=f(x) by saying that an infinitesimal change in necessarily produces an infinitesimal change in , 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. ==Functions of a single variable==

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). -definition of limit Suppose f: \R \rightarrow \R is a function defined on the real line, and there are two real numbers and . One would say: "The limit of of , as approaches , exists, and it equals ". and write, \lim_{x \to p} f(x) = L, or alternatively, say " tends to as tends to ", and write, f(x) \to L \text{ as } x \to p, if the following property holds: for every real , there exists a real such that for all real , implies . Let f : S \to \R be a real-valued function defined on some S \subseteq \R. Let be a limit point of some T \subset S—that is, is the limit of some sequence of elements of distinct from . Then we say the limit of , as approaches from values in , is , 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 Note, can be any subset of , the domain of . And the limit might depend on the selection of . This generalization includes as special cases limits on an interval, as well as left-handed limits of real-valued functions (e.g., by taking to be an open interval of the form ), and right-handed limits (e.g., by taking to be an open interval of the form ). 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 approaches 0 from above: \lim_{ {x\to 0} \atop {x\in [0, \infty)} } \sqrt{x} = 0 since for every , we may take such that for all , if , then . This definition allows a limit to be defined at limit points of the domain , if a suitable subset 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 . 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 . Deleted versus non-deleted limits The definition of limit given here does not depend on how (or whether) is defined at . Bartle refers to this as a deleted limit, because it excludes the value of at . The corresponding non-deleted limit does depend on the value of at , if is in the domain of . Let f : S \to \R be a real-valued function. The non-deleted limit of , as approaches , is if (\forall \varepsilon > 0 )\, (\exists \delta > 0) \, (\forall x \in S)\, (|x - p| The definition is the same, except that the neighborhood now includes the point , in contrast to the deleted neighborhood . 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 Examples Non-existence of one-sided limit(s) The function f(x)=\begin{cases} \sin\frac{5}{x-1} & \text{ for } x1 \end{cases} has no limit at (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 -coordinate. The function 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 -coordinate. Non-equality of one-sided limits The function f(x)=\begin{cases} 1 & \text{ for } x has a limit at every non-zero -coordinate (the limit equals 1 for negative and equals 2 for positive ). The limit at does not exist (the left-hand limit equals 1, whereas the right-hand limit equals 2). Limits at only one point The functions f(x)=\begin{cases} x & x \text{ rational } \\ 0 & x \text{ irrational } \end{cases} and f(x)=\begin{cases} 0 & x \text{ irrational } \end{cases} both have a limit at and it equals 0. Limits at countably many points The function f(x)=\begin{cases} \sin x & x \text{ irrational } \\ 1 & x \text{ rational } \end{cases} has a limit at any -coordinate of the form \tfrac{\pi}{2} + 2n\pi, where is any integer. ==Limits involving infinity==

Limits at infinity Let f:S \to \R be a function defined on S \subseteq \R. The limit of as approaches infinity is , 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| Similarly, the limit of as approaches minus infinity is , denoted \lim_{x \to -\infty}f(x) = L, means that: (\forall \varepsilon > 0)\, (\exists c > 0) \,(\forall x \in S)\, (x For example, \lim_{x \to \infty} \left(-\frac{3\sin x}{x} + 4\right) = 4 because for every , we can take such that for all real , if , then . Another example is that \lim_{x \to -\infty}e^{x} = 0 because for every , we can take {{math|1=c = max{1, −ln(ε)} }} such that for all real , if , then . 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 f:S \to\mathbb{R} be a function defined on S\subseteq\mathbb{R}. The statement the limit of as approaches is infinity, denoted \lim_{x \to p} f(x) = \infty, means that: (\forall N > 0)\, (\exists \delta > 0)\, (\forall x \in S)\, (0 N) . The statement the limit of as approaches is minus infinity, denoted \lim_{x \to p} f(x) = -\infty, means that: (\forall N > 0) \, (\exists \delta > 0) \, (\forall x \in S)\, (0 For example, \lim_{x \to 1} \frac{1}{(x-1)^2} = \infty because for every , we can take \delta = \tfrac{1}{\sqrt{N}\delta} = \tfrac{1}{\sqrt N} such that for all real , if , then . These ideas can be used together to produce definitions for different combinations, such as \lim_{x \to \infty} f(x) = \infty, or \lim_{x \to p^+}f(x) = -\infty. For example, \lim_{x \to 0^+} \ln x = -\infty because for every , we can take such that for all real , if , then . 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 for some • a neighborhood of ∞ is defined to contain an interval where and • a neighborhood of is defined in the normal way metric space In this case, is a topological space and any function of the form f : X \to Y with X, Y \subseteq \overline \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. Alternative notation 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 {{tmath|\R \cup \{-\infty, +\infty\} }} and the projectively extended real line is {{tmath|\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. Limits at infinity for rational functions There are three basic rules for evaluating limits at infinity for a rational function f(x) = \tfrac{p(x)}{q(x)} (where and are polynomials): • If the degree of is greater than the degree of , then the limit is positive or negative infinity depending on the signs of the leading coefficients; • If the degree of and are equal, the limit is the leading coefficient of divided by the leading coefficient of ; • If the degree of is less than the degree of , the limit is 0. If the limit at infinity exists, it represents a horizontal asymptote at . 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 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 as approaches is , written \lim_{(x,y) \to (p, q)} f(x, y) = L if the following condition holds: :For every , there exists a such that for all in and in , whenever 0 we have , or formally: (\forall \varepsilon > 0)\, (\exists \delta > 0)\, (\forall x \in S) \, (\forall y \in T)\, (0 Here \sqrt{(x-p)^2 + (y-q)^2} is the Euclidean distance between and . (This can in fact be replaced by any norm , 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 , we can take \delta = \sqrt \varepsilon such that for all real and real , if 0 then . Similar to the case in single variable, the value of at does not matter in this definition of limit. For such a multivariable limit to exist, this definition requires the value of approaches along every possible path approaching . In the above example, the function 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\theta, r\sin\theta) \to (0, 0), which gives \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 is a function of r which controls the shape of the path along which is approaching . Since is bounded between [−1, 1], by the sandwich theorem, this limit tends to 0. In contrast, the function f(x, y) = \frac{xy}{x^2 + y^2} does not have a limit at . Taking the path , we obtain \lim_{t \to 0} f(t, 0) = \lim_{t \to 0} \frac{0}{t^2} = 0, while taking the path , we obtain \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, does not tend to a single value as approaches . 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. Let f : S \times T \to \R be defined on S \times T \subseteq \R^2, we say the double limit of as approaches and approaches is , 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 For such a double limit to exist, this definition requires the value of approaches along every possible path approaching , excluding the two lines and . As a result, the multiple limit is a weaker notion than the ordinary limit: if the ordinary limit exists and equals , then the multiple limit exists and also equals . 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 is restricted to (S\setminus\{p\}) \times (T\setminus\{q\}), then the two definitions of limits coincide. ==Functions on metric spaces==

Suppose and are subsets of metric spaces and , respectively, and is defined between and , with , a limit point of and . It is said that the limit of as approaches is and write \lim_{x \to p}f(x) = L if the following property holds: (\forall \varepsilon > 0 )\, (\exists \delta > 0) \,(\forall x \in M) \,(0 Again, note that need not be in the domain of , nor does need to be in the range of , and even if is defined it need not be equal to . Euclidean metric The limit in Euclidean space is a direct generalization of limits to vector-valued functions. For example, we may consider a function f:S \times T \to \R^3 such that f(x, y) = (f_1(x, y), f_2(x, y), f_3(x, y) ). Then, under the usual Euclidean metric, \lim_{(x, y) \to (p, q)} f(x, y) = (L_1, L_2, L_3) if the following holds: {{block indent|For every , there exists a such that for all in and in , 0 implies \sqrt{(f_1-L_1)^2 + (f_2-L_2)^2 + (f_3-L_3)^2} }} (\forall \varepsilon > 0 )\, (\exists \delta > 0) \, (\forall x \in S) \, (\forall y \in T)\, \left(0 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: Then, we will say the uniform limit of on as approaches is and write \underset{ {x \to p} \atop {y \in T} }{\mathrm{unif} \lim \;} f(x, y) = g(y), or \lim_{x \to p}f(x, y) = g(y) \;\; \text{uniformly on} \; T, if the following holds: {{block indent|For every , there exists a such that for all in , implies \sup_{y \in T}|f(x,y) - g(y)| }} (\forall \varepsilon > 0 )\, (\exists \delta > 0) \,(\forall x \in S) \,(0 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 X and Y are topological spaces with Y a Hausdorff space. Let p be a limit point of \Omega\subseteq X, and L\in Y. For a function f: \Omega \to Y, it is said that the limit of f as x approaches p is L, written :\lim_{x \to p}f(x) = L , if the following property holds: :for every open neighborhood V of L, there exists an open neighborhood U of p such that f(U\cap \Omega-\{p\})\subseteq V. This last part of the definition can also be phrased as "there exists an open punctured neighbourhood U of p such that f(U\cap\Omega)\subseteq V. The domain of f does not need to contain p. If it does, then the value of f at p is irrelevant to the definition of the limit. In particular, if the domain of f is X\setminus\{p\} (or all of X), then the limit of f as x\to p exists and is equal to if, for all subsets of with limit point p, the limit of the restriction of f to exists and is equal to . Sometimes this criterion is used to establish the non-existence of the two-sided limit of a function on 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 Y be a Hausdorff space can be relaxed to the assumption that Y 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 p of and in its domain if and only if f(p) is the (or, in the general case, a) limit of f(x) as x tends to p. There is another type of limit of a function, namely the sequential limit. Let f:X\to Y be a mapping from a topological space into a Hausdorff space , p\in X a limit point of and . The sequential limit of f as x tends to p is if :For every sequence (x_n) in X\setminus\{p\} that converges to p, the sequence f(x_n) converges to . If is the limit (in the sense above) of f as x approaches p, then it is a sequential limit as well; however, the converse need not hold in general. If in addition is metrizable, then is the sequential limit of f as x approaches p if and only if it is the limit (in the sense above) of f as x approaches p. == 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: \lim_{x\to a}f(x)=L if, and only if, for all sequences (with, for all , not equal to ) converging to the sequence converges to . 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 to converge to 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 be a real-valued function with the domain . Let be the limit of a sequence of elements of {{math|Dm(f ) \ {a}.}} Then the limit (in this sense) of is as approaches if for every sequence {{math|x ∈ Dm(f ) \ {a} }} (so that for all , is not equal to ) that converges to , the sequence converges to . This is the same as the definition of a sequential limit in the preceding section obtained by regarding the subset of as a metric space with the induced metric. In non-standard calculus In non-standard calculus the limit of a function is defined by: \lim_{x\to a}f(x)=L if and only if for all x\in \R^*, f^*(x)-L is infinitesimal whenever is infinitesimal. Here \R^* are the hyperreal numbers and is the natural extension of to the non-standard real numbers. Keisler proved that such a hyperreal definition of limit reduces the quantifier complexity by two quantifiers. 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. 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". 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". A point is defined to be near a set A\subseteq \R if for every there is a point so that . In this setting the \lim_{x\to a} f(x)=L if and only if for all A\subseteq \R, is near whenever is near . Here is the set \{f(x) | x \in A\}. This definition can also be extended to metric and topological spaces. ==Relationship to continuity==

The notion of the limit of a function is very closely related to the concept of continuity. A function is said to be continuous at if it is both defined at and its value at equals the limit of as approaches : \lim_{x\to c} f(x) = f(c). We have here assumed that is a limit point of the domain of . ==Properties==

If a function is real-valued, then the limit of at is if and only if both the right-handed limit and left-handed limit of at exist and are equal to . The function is continuous at if and only if the limit of as approaches exists and is equal to . If is a function between metric spaces and , then it is equivalent that transforms every sequence in which converges towards into a sequence in which converges towards . If is a normed vector space, then the limit operation is linear in the following sense: if the limit of as approaches is and the limit of as approaches is , then the limit of as approaches is . If is a scalar from the base field, then the limit of as approaches is . If and are real-valued (or complex-valued) functions, then taking the limit of an operation on and (e.g., , , , , ) under certain conditions is compatible with the operation of limits of and . 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). \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} These rules are also valid for one-sided limits, including when 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. \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} \\[4pt] q^\infty & = & \begin{cases} 0 & \text{if } 0 1 \end{cases} \\[4pt] q^{-\infty} & = & \begin{cases} \infty & \text{if } 0 1 \end{cases} \end{array} (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 and . These indeterminate forms are: \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} See further L'Hôpital's rule below and Indeterminate form. Limits of compositions of functions In general, from knowing that \lim_{y \to b} f(y) = c and \lim_{x \to a} g(x) = b, it does not follow that \lim_{x \to a} f(g(x)) = c. However, this "chain rule" does hold if one of the following additional conditions holds: • (that is, is continuous at ), or • does not take the value near (that is, there exists a such that if then ). As an example of this phenomenon, consider the following function that violates both additional restrictions: f(x) = g(x) = \begin{cases} 0 & \text{if } x\neq 0 \\ 1 & \text{if } x=0 \end{cases} Since the value at is a removable discontinuity, \lim_{x \to a} f(x) = 0 for all . Thus, the naïve chain rule would suggest that the limit of is 0. However, it is the case that f(f(x))=\begin{cases} 1 & \text{if } x\neq 0 \\ 0 & \text{if } x = 0 \end{cases} and so \lim_{x \to a} f(f(x)) = 1 for all . Limits of special interest Rational functions For a nonnegative integer and constants a_1, a_2, a_3,\ldots, a_n and b_1, b_2, b_3,\ldots, b_n, \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} This can be proven by dividing both the numerator and denominator by . 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 \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} Exponential functions \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} Logarithmic functions \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} L'Hôpital's rule This rule uses derivatives to find limits of indeterminate forms or , and only applies to such cases. Other indeterminate forms may be manipulated into this form. Given two functions and , defined over an open interval containing the desired limit point , then if: • \lim_{x \to c}f(x)=\lim_{x \to c}g(x)=0, or \lim_{x \to c}f(x)=\pm\lim_{x \to c}g(x) = \pm\infty, and • f and g are differentiable over I \setminus \{c\}, and • g'(x)\neq 0 for all x \in I \setminus \{c\}, and • \lim_{x\to c}\tfrac{f'(x)}{g'(x)} exists, then: \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: \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}. 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 \lim_{n \to \infty} \sum_{i=s}^n f(i) is \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 \lim_{x \to \infty} \int_a^x f(t) \; dt is \int_a^\infty f(t) \; dt. A short way to write the limit \lim_{x \to -\infty} \int_x^b f(t) \; dt is \int_{-\infty}^b f(t) \; dt. ==See also==