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Unformatted text preview: δ )  f ( x ) – 3 < ε ” game? ..?. . Adam wins, by taking = 1 (or anything smaller). Moral: lim x → a f ( x ) need not equal f ( a ). Another example: Let f ( x ) = x +1 for all x ≠ 1, with f (1) = 3. [Draw a picture.] What is lim x → 1 f ( x )? ..?. . lim x → 1 f ( x ) = 2, even though f (1) = 3. Who wins the “(for all > 0) (there exists > 0 such that) (for all x with 0 <  x –1 < )  f ( x ) – 2 < ” game? ..?. . Eve wins. What’s Eve’s strategy for winning the game? ..?. . = . Check: 0 <  x –1 < implies  f ( x ) – 2 =  x +1 – 2 =  x –1 < = . This verifies the claim “(for all > 0) (there exists > 0 such that) (for all x with 0 <  x –1 < )  f ( x ) – 2 < ”....
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 Fall '11
 Staff
 Calculus, Limit, Adam, Eve, Adam and Eve, Garden of Eden

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