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Unformatted text preview: Math 1a: Limit Laws (Solutions) September 16, 2009 1. Let f be a function such that f (2 . 01) = 0, f (2 . 001) = 0, f (2 . 0001) = 0, and so on. Can we conclude that lim x 2 f ( x ) = 0? Solution. No, we cannot. To conclude that lim x 2 f ( x ) = 0, we must know that for any x sufficiently close to 2, f ( x ) is close to 0 and not just for x of the form 2 . 01, 2 . 001, 2 . 0001 etc. 2. Suppose we know that lim x 5 f ( x ) = 3. Which of the following must be true? 1. f (5) = 3; 2. f is defined at 5; 3. if f is defined at 5 then f (5) = 3; 4. there is a number x such that f ( x ) < 3 . 05; (hint: look for x near 5) 5. there is a number x such that f ( x ) > 2 . 95; Solution. (a), (b), (c) are false. In general, in dealing with lim x a f ( x ), the behavior of f at a is irrelevant. (d) and (e) are true. Since lim x 5 f ( x ) = 3, if we take any x sufficiently close to 5, then f ( x ) will be within distance 0 . 05 of 3. That is, for an x sufficiently close to 5 (but not equal to 5), we have 2...
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This note was uploaded on 03/27/2012 for the course MATH 1a taught by Professor Benedictgross during the Fall '09 term at Harvard.
- Fall '09