Ans4 - MAA 4200 Homework Solutions Sequential Limit Theorem = Theorem 2.1 Algebra of Limits Theorem = Theorem 2.4[2.12 Prove that if limxx0 f(x = L

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
MAA 4200: Homework Solutions June 19, 2009 Sequential Limit Theorem = Theorem 2 . 1 Algebra of Limits Theorem = Theorem 2 . 4 [2.12] Prove that if lim x x 0 f ( x ) = L then lim x x 0 | f ( x ) | = | L | . Preliminary Work: || f ( x ) | - | L || ≤ | f ( x ) - L | for all x D by the triangle inequality. Proof: Let ± > 0. By definition of limit, there exists δ 1 > 0 such that | f ( x ) - L | < ± for 0 < | x - x 0 | < δ 1 . Choose δ = δ 1 and let 0 < | x - x 0 | < δ . Then || f ( x ) | - | L || ≤ | f ( x ) - L | < ± . [2.13] In Example 2.6 on p. 71 it is proved that lim x x 0 [ x ] exists iff x 0 is not an integer. Thus, since lim x x 0 exists for all x 0 R , the limit lim x x 0 ( x - [ x ]) exists for all x 0 / Z by the Algebra of Limits Theorem. We can also apply the Algebra of Limits Theorem to show that lim x x 0 ( x - [ x 0 ]) does not exist for x 0 Z . Let f ( x ) = x - [ x ]. Proof by contradiction: suppose lim x x 0 f ( x ) exists for some x 0 Z . Then lim x x 0 [ x ] = lim x x 0 ( x - f ( x )) exists by the Algebra of Limits
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 08/13/2009 for the course MAT 371 taught by Professor Thieme during the Spring '07 term at ASU.

Page1 / 2

Ans4 - MAA 4200 Homework Solutions Sequential Limit Theorem = Theorem 2.1 Algebra of Limits Theorem = Theorem 2.4[2.12 Prove that if limxx0 f(x = L

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online