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09.19 - [Collect summaries of section 1.4[Hand out...

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[Collect summaries of section 1.4] [Hand out time-sheets for assignment #3] Section 1.3: The notion of limit (continued) Claim (see Example 1 from p. 24): If A ( h ) = 4.9(10+ h ) for all h 0, lim h 0 A ( h ) = 49. Bad proof: “Just plug in h = 0.” What’s wrong with it is that, in general, lim x a f ( x ) need not equal f ( a ). Later on, we’ll prove general theorems that guarantee that in certain fairly broad situations, lim x a f ( x ) = f ( a ). When we can appeal to one of those theorems, we get to say “Since f ( x ) is continuous at a , we can just plug in x = a and that’ll count as a valid proof. But we’re not there yet; so for now we have get down and dirty with epsilons and deltas. To show that lim h 0 A ( h ) = 49 is TRUE, we need to show that for every ε > 0 there is a number δ > 0 such that if 0 < | h | < δ , | A ( h ) – 49| < ε . To figure out what δ we need (which will depend on ε ), let’s rewrite | A ( h ) – 49|: since
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A ( h ) = 4.9(10+ h ), we have | A ( h ) – 49| = |4.9(10+ h ) – 49| = |49 + 4.9 h – 49| = |4.9 h | = 4.9 | h |. So, we need to show that for every ε > 0 there is a number δ
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