MthT 430 Chapter 5 Limits - An Equivalent Definition

This note is an attempt to clear up the confusion I (JL) probably created at the beginning of class October 10, 2007.

It is also closely related to problems 5.25 and 5.26 in Spivak.

In applying the definition of lim_{x ® a} f(x) = L, it is sometimes convenient to multiply e and/or d by positive constants.

Limits - Definitions

Definition (Actual, p. 96). 

lim x ® a  f(x) = L

means: For every e > 0, there is some d > 0 such that, for all x, if 0 < |x -a| < d, then |f(x) - L| < e.

An Equivalent Definition of Limit

Definition 10⁶. 

lim x ® a  f(x) = L

means: For every e > 0, there is some d > 0 such that, for all x, if 0 < |x -a| < 10⁶d, then |f(x) - L| < e.

10⁶ Þ Actual: Fix e > 0. By 10⁶, there is a d > 0 such that, for all x, if 0 < |x -a| < 10⁶d, then |f(x) - L| < e. If 0 < |x -a| < d, then 0 < |x -a| < 10⁶d, so |f(x) - L| < e.

Actual Þ 10⁶: Fix e > 0. By Actual, there is an h > 0 such that, for all x, if 0 < |x -a| < h, then |f(x) - L| < e. Let d = 10^-6h. Then d > 0 iff h > 0. For all x, if 0 < |x - a| < 10⁶d = h, |f(x) - L| < e.

10⁶ Þ Actual: (another Proof) Fix e > 0. By 10⁶, there is a r > 0 such that, for all x, if 0 < |x -a| < 10⁶r, then |f(x) - L| < e. Let d = 10⁶r. Then d > 0 iff r > 0. For all x, if 0 < |x - a| < d = 10⁶ r, |f(x) - L| < e.

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On 11 Oct 2007, 09:56.