limcont - Limits and Continuity In calculus, we often ask...

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L imits and C ontinuity In calculus, we often ask what value a function is approaching for a given x value. Thus, we want to know what the limit of the function is as we approach that x -value. Formally, a limit is defined as follows: Definition of a Limit: Let f(x) be a function defined on an interval that contains xa = . Then, lim ( ) f xL = if for every 0 ε > , there exists a 0 δ > such that () fx L −< whenever < When I first saw the above definition, I didn’t know what it meant. It took me a while to fully understand what it means. The definition says supposing the limit exists, then we can set the distance between the function and the limit small (less than ), by finding a corresponding that will make it happen. A picture really helps to explain this. Assuming the limit, L , exists, we can draw two horizontal lines, one at L and the other at L + . Now we can draw two vertical lines, one at a and the other at a + . a L a L a −δ L −ε Any x -value in the pink area will be closer to a than either a or a + . This
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This note was uploaded on 08/31/2011 for the course MATH 20A taught by Professor Staff during the Fall '08 term at UCSD.

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limcont - Limits and Continuity In calculus, we often ask...

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