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12 to compute limits rigorously. These computations prove that the “epsilon - delta” conditions hold. 3 1. Drill with Rules of Infinitesimal, Finite and Infinite Numbers In the following formulas, 0<ε≈0 and a ≈ 2, 0 < δ ≈ 0, b ≈ 5, c ≈ −7, H and but K a = 2, are infinite and positive. ) 1 y =ε×δ 2 y =ε−δ 3 y = ε/b 4 y = ε/δ 5 y= a+7ε b−4δ 6 y = b/ε 7 y =a+b−c 8 y =a+δ 9 y =c−ε 10 y =a−2 13 y= 16 11 y= 1 a−2 12 y= 1 a−b c a−b 14 y= 2−δ a 15 y= 5δ 4 −3δ 2 +2δ δ y= 1 H 17 y= 2−δ a−K 18 y= 5δ 4 −3δ 2 +2δ 4δ+δ 2 19 y= H 2 +3H H 20 y= H 2 +3H H2 21 y= 3δ 2 δ+8δ 2 22 y= H−K H 23 y= H−K HK 24 y= H−K H+K 25 y= √ H 26 y= √ δ 27 y= H+K H−K 28 y= √ H H+a 29 y= √ √ a+δ− a 30 y= 1 b+δ − 1 b 42 31 3.

Proof: Since f [x] is continuous at every point of [a, b], if ξ ≈ c for a ≤ c ≤ b, then f [ξ] ≈ f [c]. 11 c lies in the interval and x ≈ c. Let x1 and x2 be any two points in [a, b] with x1 ≈ x2 . ) We have f [x1 ] ≈ f [c] ≈ f [x2 ] so for any numbers x1 ≈ x2 in [a, b], f [x1 ] ≈ f [x2 ]. Suppose the conclusion of the theorem is false. Then there is a real θ > 0 such that for every γ > 0 there exist x1 and x2 in [a, b] with |x1 − x2 | < γ and |f [x1 ] − f [x2 ]| ≥ θ. 1 to this implication and select a positive infinitesimal γ ≈ 0.

1 to see that the same implication holds in the hyperreals. Moreover, x = ξ and nonzero ∆x = δx ≈ 0 satisfy the left hand side of the implication, so the right side holds. Since θ was arbitrary, condition (a) is proved. 3. Computing Locally Uniform Limits 36 3. The Theory of Limits The following limit is uniform on compact subintervals of (−∞, ∞). 12. The difference is infinitesimal (3 x2 + 3 x δx + δx2 ) − 3 x2 = (3 x + δx)δx when δx is infinitesimal. First, 3 x+δx is finite because a sum of finite numbers is finite.

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Mathematical Background. Foundations of Infinitesimal Calculus by K.D.Stroyan

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