Imagine a and b located one meter apart on the x axis

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Unformatted text preview: ference ∆ = C a−x C as x → a = 0.5, be it from the left or from the right. left right ∆ = C a−x C = 0.01, 0.001, 0.0001, 0.00001, . . . After a while it becomes infinitely small and we denote it by δ. We may say : infinitel infinitely 1 δ = /10n as n → ∞ 1 We can see that LIMIT { /10n } = 0 . n→∞ Let us look at another example. Imagine A and B located one meter apart on the x-axis . At each second let B jump 1/2 distance towards A. We may say that B → A. / / A ← 18 14 a 1 /2 B x-axis a + 1 meter The difference ∆ = C A − BC = 1 2 , 1 4 , 1 8 , . . . /// After a while it becomes infinitely small and we denote it by δ. We may say : infinitel infinitely 1 δ = /2n as n → ∞ 1 We can see that LIMIT { /2n } = 0 . n→∞ By definition: An infinitely small q uantity whose LIMIT is zero is infinitely called an INFINITESIMAL. Note that the LIMIT is zero, i.e. the quantity δ TENDS TO zero but T ENDS does not become z er o. δ does not v anish. zer ero vanish. d oes 16 We may now use the infinitesimal δ and say: x → a+ ≡ x = a + δ x → a-- ≡ x = a − δ B→A ≡ B=a+δ To express the concept x coincides with a, we may say : from the right : x = Limit { a + δ } right δ→0 from the left : x = Limit { a − δ } l eft δ→0 Likewise, to express B coincides with A we may say : B = Limit {a+δ} δ→0 As B jumps from 1 to 1 2 to 1 4 to 1 8 . . . to A it skips a lot points in between. / / / points Likewise, as x → a it skips a lot of points in between. This is because δ is a points discrete infinitesimal . We can think of the number line or x-axis as smooth and CONTINUOUS.There are no gaps or breaks. In Calculus we want to study the behavior of a function at each and every point on the x-axis or instant on the TIME AXIS. So we want x to TEND TO a point in a smooth and CONTINUOUS manner. For the infinitesimal that TENDS TO zero in a smooth and CONTINUOUS manner we have the special notation δ x. Geometrically, we can think of δ x as a small CONTINUOUS line segment representing an infinitely small CHANGE in x. The length of this small line segment δ x is always > 0 no matter where we are on the x-axis. δ x gets smaller and smaller. The length of δ x TENDS TO zero but does not become zero. δ x does not vanish. However, the LIMIT of δx is zero. 17 We may let : 1 δx = /10x as x → ∞ 1 or δx = /2x as x → ∞ Now we may express x → a in a continuous manner by : continuous x → a+ ≡ x = a + δx x → a-...
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This note was uploaded on 11/29/2012 for the course PHYSICS 105 taught by Professor Tamerdoğan during the Fall '09 term at Middle East Technical University.

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