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41-721: (Redirected from Equivalent ) [REDACTED] The present page holds the title of a primary topic , and an article needs to be written about it. It is believed to qualify as a broad-concept article . It may be written directly at this page or drafted elsewhere and then moved to this title. Related titles should be described in Equivalence , while unrelated titles should be moved to Equivalence (disambiguation) . ( March 2018 ) [REDACTED] [REDACTED] Look up equivalence  or equivalent in Wiktionary,

82-563: A National Collegiate Athletic Association concept Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title Equivalence . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Equivalence&oldid=1186031519 " Category : Disambiguation pages Hidden categories: Disambiguation pages to be converted to broad concept articles Short description

123-563: A National Collegiate Athletic Association concept Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title Equivalence . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Equivalence&oldid=1186031519 " Category : Disambiguation pages Hidden categories: Disambiguation pages to be converted to broad concept articles Short description

164-599: A principle related to risk premium Economic equivalence , a concept in engineering economics Ricardian equivalence , or Ricardo–de Viti–Barro equivalence, a proposition in economics Mathematics [ edit ] Equality (mathematics) Equivalence relation Equivalence class Equivalence of categories , in category theory Equivalent infinitesimal Identity Matrix equivalence in linear algebra Turing equivalence (recursion theory) Elementary equivalence , in mathematical logic Physics [ edit ] Equivalence principle in

205-599: A principle related to risk premium Economic equivalence , a concept in engineering economics Ricardian equivalence , or Ricardo–de Viti–Barro equivalence, a proposition in economics Mathematics [ edit ] Equality (mathematics) Equivalence relation Equivalence class Equivalence of categories , in category theory Equivalent infinitesimal Identity Matrix equivalence in linear algebra Turing equivalence (recursion theory) Elementary equivalence , in mathematical logic Physics [ edit ] Equivalence principle in

246-572: A wide variety of uncommon others, where each expression stands for the limit of a function constructed by an arithmetical combination of two functions whose limits respectively tend to ⁠ 0 , {\displaystyle 0,} ⁠ ⁠ 1 , {\displaystyle 1,} ⁠ or ⁠ ∞ {\displaystyle \infty } ⁠ as indicated. A limit taking one of these indeterminate forms might tend to zero, might tend to any finite value, might tend to infinity, or might diverge, depending on

287-1048: Is a brief proof: Suppose there are two equivalent infinitesimals α ∼ α ′ {\displaystyle \alpha \sim \alpha '} and β ∼ β ′ {\displaystyle \beta \sim \beta '} . lim β α = lim β β ′ α ′ β ′ α ′ α = lim β β ′ lim α ′ α lim β ′ α ′ = lim β ′ α ′ {\displaystyle \lim {\frac {\beta }{\alpha }}=\lim {\frac {\beta \beta '\alpha '}{\beta '\alpha '\alpha }}=\lim {\frac {\beta }{\beta '}}\lim {\frac {\alpha '}{\alpha }}\lim {\frac {\beta '}{\alpha '}}=\lim {\frac {\beta '}{\alpha '}}} For

328-772: Is different from Wikidata All article disambiguation pages All disambiguation pages Equivalence (disambiguation) (Redirected from Equivalence (disambiguation) ) [REDACTED] The present page holds the title of a primary topic , and an article needs to be written about it. It is believed to qualify as a broad-concept article . It may be written directly at this page or drafted elsewhere and then moved to this title. Related titles should be described in Equivalence , while unrelated titles should be moved to Equivalence (disambiguation) . ( March 2018 ) [REDACTED] [REDACTED] Look up equivalence  or equivalent in Wiktionary,

369-960: Is different from Wikidata All article disambiguation pages All disambiguation pages Equivalent infinitesimal lim x → c ( f ( x ) + g ( x ) ) = lim x → c f ( x ) + lim x → c g ( x ) , lim x → c ( f ( x ) g ( x ) ) = lim x → c f ( x ) ⋅ lim x → c g ( x ) , {\displaystyle {\begin{aligned}\lim _{x\to c}{\bigl (}f(x)+g(x){\bigr )}&=\lim _{x\to c}f(x)+\lim _{x\to c}g(x),\\[3mu]\lim _{x\to c}{\bigl (}f(x)g(x){\bigr )}&=\lim _{x\to c}f(x)\cdot \lim _{x\to c}g(x),\end{aligned}}} and likewise for other arithmetic operations; this

410-434: Is insufficient to determinate the limit An expression that arises by ways other than applying the algebraic limit theorem may have the same form of an indeterminate form. However it is not appropriate to call an expression "indeterminate form" if the expression is made outside the context of determining limits. An example is the expression 0 0 {\displaystyle 0^{0}} . Whether this expression

451-447: Is left undefined, or is defined to equal 1 {\displaystyle 1} , depends on the field of application and may vary between authors. For more, see the article Zero to the power of zero . Note that 0 ∞ {\displaystyle 0^{\infty }} and other expressions involving infinity are not indeterminate forms . The indeterminate form 0 / 0 {\displaystyle 0/0}

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492-675: Is not an indeterminate form. The expression 0 + ∞ {\displaystyle 0^{+\infty }} obtained from considering lim x → c f ( x ) g ( x ) {\displaystyle \lim _{x\to c}f(x)^{g(x)}} gives the limit 0 , {\displaystyle 0,} provided that f ( x ) {\displaystyle f(x)} remains nonnegative as x {\displaystyle x} approaches c {\displaystyle c} . The expression 0 − ∞ {\displaystyle 0^{-\infty }}

533-582: Is not commonly regarded as an indeterminate form, because if the limit of f / g {\displaystyle f/g} exists then there is no ambiguity as to its value, as it always diverges. Specifically, if f {\displaystyle f} approaches 1 {\displaystyle 1} and g {\displaystyle g} approaches 0 , {\displaystyle 0,} then f {\displaystyle f} and g {\displaystyle g} may be chosen so that: In each case

574-417: Is not sufficient to evaluate the limit lim x → c f ( x ) g ( x ) . {\displaystyle \lim _{x\to c}f(x)^{g(x)}.} If the functions f {\displaystyle f} and g {\displaystyle g} are analytic at c {\displaystyle c} , and f {\displaystyle f}

615-518: Is particularly common in calculus , because it often arises in the evaluation of derivatives using their definition in terms of limit. As mentioned above, while This is enough to show that 0 / 0 {\displaystyle 0/0} is an indeterminate form. Other examples with this indeterminate form include and Direct substitution of the number that x {\displaystyle x} approaches into any of these expressions shows that these are examples correspond to

656-468: Is positive for x {\displaystyle x} sufficiently close (but not equal) to c {\displaystyle c} , then the limit of f ( x ) g ( x ) {\displaystyle f(x)^{g(x)}} will be 1 {\displaystyle 1} . Otherwise, use the transformation in the table below to evaluate the limit. The expression 1 / 0 {\displaystyle 1/0}

697-872: Is similarly equivalent to 1 / 0 {\displaystyle 1/0} ; if f ( x ) > 0 {\displaystyle f(x)>0} as x {\displaystyle x} approaches c {\displaystyle c} , the limit comes out as + ∞ {\displaystyle +\infty } . To see why, let L = lim x → c f ( x ) g ( x ) , {\displaystyle L=\lim _{x\to c}f(x)^{g(x)},} where lim x → c f ( x ) = 0 , {\displaystyle \lim _{x\to c}{f(x)}=0,} and lim x → c g ( x ) = ∞ . {\displaystyle \lim _{x\to c}{g(x)}=\infty .} By taking

738-851: Is sometimes called the algebraic limit theorem . However, certain combinations of particular limiting values cannot be computed in this way, and knowing the limit of each function separately does not suffice to determine the limit of the combination. In these particular situations, the limit is said to take an indeterminate form , described by one of the informal expressions 0 0 ,   ∞ ∞ ,   0 × ∞ ,   ∞ − ∞ ,   0 0 ,   1 ∞ ,  or  ∞ 0 , {\displaystyle {\frac {0}{0}},~{\frac {\infty }{\infty }},~0\times \infty ,~\infty -\infty ,~0^{0},~1^{\infty },{\text{ or }}\infty ^{0},} among

779-723: Is the quotient of two functions each of which converges to zero. This indeterminate form is denoted by 0 / 0 {\displaystyle 0/0} . For example, as x {\displaystyle x} approaches 0 , {\displaystyle 0,} the ratios x / x 3 {\displaystyle x/x^{3}} , x / x {\displaystyle x/x} , and x 2 / x {\displaystyle x^{2}/x} go to ∞ {\displaystyle \infty } , 1 {\displaystyle 1} , and 0 {\displaystyle 0} respectively. In each case, if

820-646: Is used in the 4th equality, and 1 − cos ⁡ x ∼ x 2 2 {\displaystyle 1-\cos x\sim {x^{2} \over 2}} is used in the 5th equality. L'Hôpital's rule is a general method for evaluating the indeterminate forms 0 / 0 {\displaystyle 0/0} and ∞ / ∞ {\displaystyle \infty /\infty } . This rule states that (under appropriate conditions) where f ′ {\displaystyle f'} and g ′ {\displaystyle g'} are

861-465: The derivatives of f {\displaystyle f} and g {\displaystyle g} . (Note that this rule does not apply to expressions ∞ / 0 {\displaystyle \infty /0} , 1 / 0 {\displaystyle 1/0} , and so on, as these expressions are not indeterminate forms.) These derivatives will allow one to perform algebraic simplification and eventually evaluate

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902-679: The direct effect of European Union law Logic [ edit ] Logical equivalence , where two statements are logically equivalent if they have the same logical content Material equivalence , a relationship where the truth of either one of the connected statements requires the truth of the other Science and technology [ edit ] Chemistry [ edit ] Equivalent (chemistry) Equivalence point Equivalent weight Computing [ edit ] Turing equivalence (theory of computation) , or Turing completeness Semantic equivalence in computer metadata Economics [ edit ] Certainty equivalent ,

943-679: The direct effect of European Union law Logic [ edit ] Logical equivalence , where two statements are logically equivalent if they have the same logical content Material equivalence , a relationship where the truth of either one of the connected statements requires the truth of the other Science and technology [ edit ] Chemistry [ edit ] Equivalent (chemistry) Equivalence point Equivalent weight Computing [ edit ] Turing equivalence (theory of computation) , or Turing completeness Semantic equivalence in computer metadata Economics [ edit ] Certainty equivalent ,

984-603: The natural logarithm (ln) is a continuous function ; it is irrelevant how well-behaved f {\displaystyle f} and g {\displaystyle g} may (or may not) be as long as f {\displaystyle f} is asymptotically positive. (the domain of logarithms is the set of all positive real numbers.) Although L'Hôpital's rule applies to both 0 / 0 {\displaystyle 0/0} and ∞ / ∞ {\displaystyle \infty /\infty } , one of these forms may be more useful than

1025-573: The 2nd equality, e y − 1 ∼ y {\displaystyle e^{y}-1\sim y} where y = x ln ⁡ 2 + cos ⁡ x 3 {\displaystyle y=x\ln {2+\cos x \over 3}} as y become closer to 0 is used, and y ∼ ln ⁡ ( 1 + y ) {\displaystyle y\sim \ln {(1+y)}} where y = cos ⁡ x − 1 3 {\displaystyle y={{\cos x-1} \over 3}}

1066-526: The absolute value | f / g | {\displaystyle |f/g|} approaches + ∞ {\displaystyle +\infty } , and so the quotient f / g {\displaystyle f/g} must diverge, in the sense of the extended real numbers (in the framework of the projectively extended real line , the limit is the unsigned infinity ∞ {\displaystyle \infty } in all three cases ). Similarly, any expression of

1107-1874: The evaluation of the indeterminate form 0 / 0 {\displaystyle 0/0} , one can make use of the following facts about equivalent infinitesimals (e.g., x ∼ sin ⁡ x {\displaystyle x\sim \sin x} if x becomes closer to zero): For example: lim x → 0 1 x 3 [ ( 2 + cos ⁡ x 3 ) x − 1 ] = lim x → 0 e x ln ⁡ 2 + cos ⁡ x 3 − 1 x 3 = lim x → 0 1 x 2 ln ⁡ 2 + cos ⁡ x 3 = lim x → 0 1 x 2 ln ⁡ ( cos ⁡ x − 1 3 + 1 ) = lim x → 0 cos ⁡ x − 1 3 x 2 = lim x → 0 − x 2 6 x 2 = − 1 6 {\displaystyle {\begin{aligned}\lim _{x\to 0}{\frac {1}{x^{3}}}\left[\left({\frac {2+\cos x}{3}}\right)^{x}-1\right]&=\lim _{x\to 0}{\frac {e^{x\ln {\frac {2+\cos x}{3}}}-1}{x^{3}}}\\&=\lim _{x\to 0}{\frac {1}{x^{2}}}\ln {\frac {2+\cos x}{3}}\\&=\lim _{x\to 0}{\frac {1}{x^{2}}}\ln \left({\frac {\cos x-1}{3}}+1\right)\\&=\lim _{x\to 0}{\frac {\cos x-1}{3x^{2}}}\\&=\lim _{x\to 0}-{\frac {x^{2}}{6x^{2}}}\\&=-{\frac {1}{6}}\end{aligned}}} In

1148-771: The expression 0 0 {\displaystyle 0^{0}} is an indeterminate form: lim x → 0 + x 0 = 1 , lim x → 0 + 0 x = 0. {\displaystyle {\begin{aligned}\lim _{x\to 0^{+}}x^{0}&=1,\\\lim _{x\to 0^{+}}0^{x}&=0.\end{aligned}}} Thus, in general, knowing that lim x → c f ( x ) = 0 {\displaystyle \textstyle \lim _{x\to c}f(x)\;=\;0} and lim x → c g ( x ) = 0 {\displaystyle \textstyle \lim _{x\to c}g(x)\;=\;0}

1189-513: The form a / 0 {\displaystyle a/0} with a ≠ 0 {\displaystyle a\neq 0} (including a = + ∞ {\displaystyle a=+\infty } and a = − ∞ {\displaystyle a=-\infty } ) is not an indeterminate form, since a quotient giving rise to such an expression will always diverge. The expression 0 ∞ {\displaystyle 0^{\infty }}

1230-609: The free dictionary. Equivalence or Equivalent may refer to: Arts and entertainment [ edit ] Album-equivalent unit , a measurement unit in the music industry Equivalence class (music) Equivalent VIII , or The Bricks , a minimalist sculpture by Carl Andre Equivalents , a series of photographs of clouds by Alfred Stieglitz Language [ edit ] Dynamic and formal equivalence in translation Equivalence (formal languages) Law [ edit ] The doctrine of equivalents in patent law The equivalence principle as if impacts on

1271-609: The free dictionary. Equivalence or Equivalent may refer to: Arts and entertainment [ edit ] Album-equivalent unit , a measurement unit in the music industry Equivalence class (music) Equivalent VIII , or The Bricks , a minimalist sculpture by Carl Andre Equivalents , a series of photographs of clouds by Alfred Stieglitz Language [ edit ] Dynamic and formal equivalence in translation Equivalence (formal languages) Law [ edit ] The doctrine of equivalents in patent law The equivalence principle as if impacts on

Equivalence - Misplaced Pages Continue

1312-430: The indeterminate form 0 / 0 {\displaystyle 0/0} , but these limits can assume many different values. Any desired value a {\displaystyle a} can be obtained for this indeterminate form as follows: The value ∞ {\displaystyle \infty } can also be obtained (in the sense of divergence to infinity): The following limits illustrate that

1353-1133: The limit does not exist, as many of the examples above show. In many cases, algebraic elimination, L'Hôpital's rule , or other methods can be used to manipulate the expression so that the limit can be evaluated. When two variables α {\displaystyle \alpha } and β {\displaystyle \beta } converge to zero at the same limit point and lim β α = 1 {\displaystyle \textstyle \lim {\frac {\beta }{\alpha }}=1} , they are called equivalent infinitesimal (equiv. α ∼ β {\displaystyle \alpha \sim \beta } ). Moreover, if variables α ′ {\displaystyle \alpha '} and β ′ {\displaystyle \beta '} are such that α ∼ α ′ {\displaystyle \alpha \sim \alpha '} and β ∼ β ′ {\displaystyle \beta \sim \beta '} , then: Here

1394-418: The limit. L'Hôpital's rule can also be applied to other indeterminate forms, using first an appropriate algebraic transformation. For example, to evaluate the form 0 : The right-hand side is of the form ∞ / ∞ {\displaystyle \infty /\infty } , so L'Hôpital's rule applies to it. Note that this equation is valid (as long as the right-hand side is defined) because

1435-479: The limits of the numerator and denominator are substituted, the resulting expression is 0 / 0 {\displaystyle 0/0} , which is indeterminate. In this sense, 0 / 0 {\displaystyle 0/0} can take on the values 0 {\displaystyle 0} , 1 {\displaystyle 1} , or ∞ {\displaystyle \infty } , by appropriate choices of functions to put in

1476-816: The natural logarithm of both sides and using lim x → c ln ⁡ f ( x ) = − ∞ , {\displaystyle \lim _{x\to c}\ln {f(x)}=-\infty ,} we get that ln ⁡ L = lim x → c ( g ( x ) × ln ⁡ f ( x ) ) = ∞ × − ∞ = − ∞ , {\displaystyle \ln L=\lim _{x\to c}({g(x)}\times \ln {f(x)})=\infty \times {-\infty }=-\infty ,} which means that L = e − ∞ = 0. {\displaystyle L={e}^{-\infty }=0.} The adjective indeterminate does not imply that

1517-720: The numerator and denominator. A pair of functions for which the limit is any particular given value may in fact be found. Even more surprising, perhaps, the quotient of the two functions may in fact diverge, and not merely diverge to infinity. For example, x sin ⁡ ( 1 / x ) / x {\displaystyle x\sin(1/x)/x} . So the fact that two functions f ( x ) {\displaystyle f(x)} and g ( x ) {\displaystyle g(x)} converge to 0 {\displaystyle 0} as x {\displaystyle x} approaches some limit point c {\displaystyle c}

1558-402: The other in a particular case (because of the possibility of algebraic simplification afterwards). One can change between these forms by transforming f / g {\displaystyle f/g} to ( 1 / g ) / ( 1 / f ) {\displaystyle (1/g)/(1/f)} . The following table lists the most common indeterminate forms and

1599-438: The specific functions involved. A limit which unambiguously tends to infinity, for instance lim x → 0 1 / x 2 = ∞ , {\textstyle \lim _{x\to 0}1/x^{2}=\infty ,} is not considered indeterminate. The term was originally introduced by Cauchy 's student Moigno in the middle of the 19th century. The most common example of an indeterminate form

1640-527: The theory of general relativity Other uses [ edit ] Equivalence (trade) Moral equivalence , a term used in political debate The Equivalent , a sum paid from England to Scotland at their Union in 1707 See also [ edit ] All pages with titles containing equivalent All pages with titles containing equivalence All pages with titles containing equivalency [REDACTED] Quotations related to Equivalence at Wikiquote ≡ (disambiguation) Equivalency,

1681-527: The theory of general relativity Other uses [ edit ] Equivalence (trade) Moral equivalence , a term used in political debate The Equivalent , a sum paid from England to Scotland at their Union in 1707 See also [ edit ] All pages with titles containing equivalent All pages with titles containing equivalence All pages with titles containing equivalency [REDACTED] Quotations related to Equivalence at Wikiquote ≡ (disambiguation) Equivalency,

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