09.21 - Section 1.3: The notion of limit (continued) For...

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Section 1.3: The notion of limit (continued) For the function f ( x ) = x sin 1/ x (defined for all x 0) do we have lim x 0 f ( x ) = 0? 0.4 0.2 0.2 0.4 0.2 0.1 0.1 0.2 0.3 0.4 ..?. . Yes. What’s Eve’s strategy? ..?. . Eve wins by taking δ = ε . Check: For every x satisfying 0 < | x – 0| < , we have | x sin 1/ x – 0| = | x sin 1/ x | = | x | |sin 1/ x | | x | 1 = | x | = | x – 0| < = . (Note that this proof makes use of the handy fact that | ab | =
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| a | | b |, which you’ll prove in the next homework assignment. Another handy fact about absolute values is that | a + b | | a | + | b |; this is called the triangle inequality .) Stewart’s paraphrase of the definition of limits: We say lim x a f ( x ) = L if the values of f ( x ) tend to get closer and closer to the number L as x gets closer and closer to the number a (from either side of a ) but x a . But what does “closer and closer” mean?
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09.21 - Section 1.3: The notion of limit (continued) For...

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