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| − | ==Limits, continuity, and convergence==
| + | Each statement of a syllogism is one of 4 types, as follows: |
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| − | Beginning perhaps with D'Alembert, it was an oft-repeated statement of 18th century mathematics that the calculus could be "based on limits." His own definition of limit is as follows:
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| − | :... the quantity to which the ratio z/u approaches more and more closely if we suppose z and u to be real and decreasing. Nothing is clearer than that.<ref>Dunham p. 72 cited in Bogomolny</ref>
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| − | | + | ! Type !! Statement !! Alternative |
| − | It is not surprising then that the arithmetization program culminated in the establishment of the concept of the limit and of those other fundamental concepts that were connected with it, namely, continuity and convergence.
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| − | | + | | style="text-align: center;" | '''A''' || '''All''' $A$ '''are''' $B$ || |
| − | ====Limits==== | + | |- |
| − | | + | | style="text-align: center;" | '''I''' || '''Some''' $A$ '''are''' $B$ || |
| − | Bolzano and Cauchy developed (independently) a concept of ''limit'' that had several advances over previous understandings:
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| − | * it was free from the ideas of motion and velocity and did not depend on geometry
| + | | style="text-align: center;" | '''E''' || '''No''' $A$ '''are''' $B$ || (= '''All''' $A$ '''are not''' $B$) |
| − | * it did not retain the (unnecessary) restriction, that a variable could never surpass its limit
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| − | Cauchy definition, in particular, stated only that the variable and its limit differed by less than any desired quantity, as follows:<ref>Grabiner (1981) p. 80</ref>
| + | | style="text-align: center;" | '''O''' || '''Not All''' $A$ '''are''' $B$ || (= '''Some''' $A$ '''are not''' $B$) |
| − | :When the successive values attributed to a variable approach indefinitely a fixed value so as to end by differing from it by as little as one wishes, this last is called the limit of all the others.
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| − | Cauchy used this definition to define the infinitesimal as a dependent variable, thus freeing it from previous understandings of it as a very small fixed number:<ref>Boyer p. 540</ref>
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| − | :One says that a variable quantity becomes infinitely small when its numerical value decreases indefinitely in such a way as to converge toward the limit zero. | |
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| − | Working with the notion of a sequence that "converges within itself," Bolzano and Cauchy sought to bind up the concepts ''limit'' and ''real number'', somewhat as follows:
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| − | :If, for a given integer $p$ and for $n$ sufficiently large, S{n+p} differs from S{n} by less than any assigned magnitude $\varepsilon$, then S{n} also converges to the (external) real number $S$, the limit of the sequence.
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| − | Meray understood the error involved in the circular way that the concepts ''limit'' and ''real number'' were defined:<ref>Boyer p. 584</ref>
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| − | * the limit (of a sequence) was defined to be a real number $S$
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| − | * a real number was defined as a limit (of a sequence of rational numbers)
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| − | Meray avoided references to convergence to an (external) real number $S$. Instead, he described convergence using only the rational numbers $n$, $p$, and $\varepsilon$, i.e. the Bolzano-Cauchy condition.
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| − | Weierstrass also understood the error involved in earlier ways of defining the concepts ''limit'' and ''irrational number'':<ref>Boyer p. 584</ref>
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| − | * the definition of the former presupposed the notion of the latter
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| − | * therefore, the the definition the latter must be independent of the former
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| − | Roughly speaking, Weierstrass defined irrational numbers not as the limit of a series, but rather as the (convergent) sequence associated with the series. His definition of limit is as follows:
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| − | :$\displaystyle \lim_{x \to \alpha}f(x) = L$ if and only if, for every $ε > 0$, there exists a $δ > 0$ so that, if $0 < |x - a| < δ$, then $|f(x) - L| < ε$.
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| − | ==Notes== | |
| − | <references/>
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| − | ==References==
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| − | * Bogomolny, A. "What Is Calculus?" from ''Interactive Mathematics Miscellany and Puzzles'' http://www.cut-the-knot.org/WhatIs/WhatIsCalculus.shtml#Alembert, Accessed 27 May 2014
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| − | * Dunham, W. (2008). ''The Calculus Gallery: Masterpieces from Newton to Lebesgue'', Princeton University Press.
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