By William McGowen Priestley (auth.)
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Extra resources for Calculus: An Historical Approach
H2 + L Imlt h 2h h-O . (h + 2) =. 2 = LImIt h-O o 22 1 Tokens from the Gods When investigating the limit of a function at a point, one may encounter any of the following situations: (I) The limit exists and agrees with the action of the function at the point. (11) The limit exists, but the function does not act accordingly. (III) The limit does not exist. If case (I) occurs, the function is said to be continuous at the point. This is illustrated in Examples 3 and 4. Case (11) is illustrated in Examples 5 and 6.
Whenever there is not agreement between action and purpose, or whenever one or both are missing, the anxieties of discontinuity emerge. At 1200, friend g, you behave discontinuously. You have a purpose : Lim g at 1200 is 0, but you do not act accordingly: g(1200) does not exist. " "Let me leave you with this idea, to ponder as you will. " 19 9. Limits Definition. A function G is said to be continuous at a point x provided that the following three conditions are satisfied: (1) G(x) exists. (2) Lim G at x exists.
The work presumes to begin from a beginning; that is, it presupposes a certain level of readiness, but makes no other prerequisites. Yet it never offers any "motivations", it has no il/uminating "asides", it does not attempt to make anything "intuitive", and it auoids applications to a fault. 1t is so "humorless" in its mathematical purism that, although it is a book about "Elements", it nevertheless does not unbend long enough in its singlemindedness to make the remark, however incidental/y, that if a rectangle has a base of 3 inches and a height of 4 inches then it has an area of 12 square inches.