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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 x a = . Then, lim ( ) x a f x L = if for every 0 ε > , there exists a 0 δ > such that ( ) f x L ε < whenever x a δ < 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 ε ), by finding a corresponding δ 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 ε and the other at L ε + . Now we can draw two vertical lines, one at a δ and the other at a δ + .
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