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Unformatted text preview: MATH 1501 Prof. C. Heil Fall 2001 LIMITS Imprecise discussion of limits . The intuitive idea of what we mean when we say lim x c f ( x ) = L is that as x approaches closer and closer to c , the function values f ( x ) approach closer and closer to L . But this is not a precise statement, it does not tell you how to figure out whether a function has a limit or what that limit is. Its like saying a hot object in a cold room will cool downfrom that statement you get no precise information about how fast the object is cooling, or when it will reach a particular temperature, etc. By contrast, Newtons law of cooling is a precise statement about how the rate of change of the temperature relates to the difference between the temperature of the object and the temperature of the surrounding medium, and this precise statement does allow you to calculate how long it will take for an object to cool down, etc. We need a similarly precise definition of the meaning of limit in order to be able to deal quantitatively with limits. Precise definition of limit . We must state EXACTLY what it means to write lim x c f ( x ) = L. The EXACT, PRECISE meaning of these symbols is that: Given any number > , there is a number > 0 such that: if 0 < | x- c | < then | f ( x )- L | < . Thus, to prove that a limit is a certain value L , we must demonstrate that given any number > 0 it is possible to find a number > 0 so that a particular something happens. Its not enough to prove that something for a single value of ; we must be able to demonstrate that no matter what is specified, there does exist a corresponding so that: if < | x- c | < then | f ( x )- L | < . Example . Let me prove that lim x 3 (5 x- 7) = 8. Im going to give the proof onlyIm not going to explain how I got the proof, I just want you to see if you can follow the proof and understand each step in the proof that I make. What we must show is that: Given any number > , there is a number > 0 such that: if 0 < |...
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