By Courant R.

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If x is nearly 1, then x + 1 is nearly 1 + 1 = 2. ” 31 32 3. 1 Plain Limits Technically, there are two equivalent ways to define the simple continuous variable limit as follows. 1. Limit Let f [x] be a real valued function defined for 0 < |x− a| < ∆ with ∆ a fixed positive real number. 2 hold. 2. Limit of a Real Variable Let f [x] be a real valued function defined for 0 < |x− a| < ∆ with ∆ a fixed positive real number. Let b be a real number. Then the following are equivalent: (a) Whenever the hyperreal number x satisfies 0 < |x − a| ≈ 0, the natural extension function satisfies f [x] ≈ b (b) For every accuracy tolerance θ there is a sufficiently small positive real number γ such that if the real number x satisfies 0 < |x − a| < γ, then |f [x] − b| < θ Proof: We show that (a) ⇒ (b) by proving that not (b) implies not (a), the contrapositive.

For example, we might have a = 2 − δ, b = 5 + 3δ can not conclude that a+b+c and c = −7 − 2δ. Then a ≈ 2, b ≈ 5 and c ≈ −7, but a + b + c = 0. (Notice that it is true 1 is not defined. ) In this case a+b+c 1 might make a + b + c = 0, so a+b+c is defined (and positive or negative infinite) in some cases, but not in others. This means that the value of 1 a+b+c can not be determined knowing only that the sum is infinitesimal. In Webster’s unabridged dictionary, the term “indeterminate” has the following symbolic characters along with the verbal definition ∞ 0 , , ∞ · 0, 1∞ , 00 , ∞0 , ∞ − ∞ 0 ∞ In the first place, Webster’s definition pre-dates the discovery of hyperreal numbers.

Let b be a real number. Then the following are equivalent: (a) Whenever the hyperreal number x satisfies 0 < |x − a| ≈ 0, the natural extension function satisfies f [x] ≈ b (b) For every accuracy tolerance θ there is a sufficiently small positive real number γ such that if the real number x satisfies 0 < |x − a| < γ, then |f [x] − b| < θ Proof: We show that (a) ⇒ (b) by proving that not (b) implies not (a), the contrapositive. Assume (b) fails. Then there is a real θ > 0 such that for every real γ > 0 there is a real x satisfying 0 < |x − a| < γ and |f [x] − b| ≥ θ.

### Advanced methods in applied mathematics, lecture course by Courant R.

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