MA 141 Study Guide - Final Guide: Asymptote, Classification Of Discontinuities, Negative Number
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Now let"s work to create a more formal de nition of a limit. We need to write a de nition for a limit such that, if f is a continuous (nice) function, lim x x0 f (x) = f (x0). Let"s say we want to restrict the values of f (x) to within a small distance from f (x0). Then we can say |f (x) f (x0)| < . This inequality will hold for the function values for any x such that f (x) lies in the blue box. However, we can choose a symmetric interval around x0 for ease of notation. Then we say we can nd a corresponding value such that |x x0| < must mean |f (x) f (x0)| < . Then, if we make smaller, so that the function values f (x) are closer to f (x0), we can nd a new such that the relationship still holds.