notes for epsillon delta proof fall 2006 MATH 125 Calculus (Jaffrey)

Notes for epsillon delta proof fall 2006 MATH 125 Calculus (Jaffrey)

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CORRECTION: Let f(x) = x 2 . Prove Lim f(x) = 4 x 2 Proof: Step 1 – We must show that for every ε-neighborhood of 4, N(4,ε), there is a δ-neighborhood about 2, N(2,δ), such that for every x in N(2,δ), excluding x = 2, then f(x) maps x into N(4,ε), i.e. y=f(x) is in N(4,ε). Step 2 – Let ε > 0 be fix arbitrarily. Step 3 – We examine when y = f(x) is in N(4,ε). (Note := if and only if) y is in N(4,ε) y satisfies | y – 4| <ε ( This is the definition of N(4,ε)) y satisfies | y – 4| <ε x satisfies |( x 2 - 4| <ε (This is follow from that y = f(x) = x 2 ) x satisfies | x 2 - 4| <ε x satisfies - ε < x 2 - 4 <ε (do not need to assume ε<4 at this point) (This follows from simplifying
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Unformatted text preview: the expression employing standard algebraic properties on inequalities) x satisfies - &lt; x 2- 4 &lt; x satisfies 4 - &lt; X 2 &lt; 4 + (More simlifications) x satisfies 4 - &lt; x 2 &lt;4 + x satisfies (4 ) &lt; x &lt;(4 + ) (here we need &lt;4) (And even more simplification of inequalities) x satisfies (4 ) &lt; x &lt;(4 + ) x satisfies (4 ) 2 &lt; x -2 &lt;(4 + ) - 2 (Should of subtracted by 2 not 4) (And even more simplification of inequalities) Step 4 Let 1 = |(4 ) 2| &amp; 1 = |(4 + ) 2| and choose = min( 1 , 2 ) Step 5 . Step 6 ....
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This note was uploaded on 09/03/2008 for the course MATH 125 taught by Professor Tuffaha during the Fall '07 term at USC.

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