MATH401 Study Guide - Final Guide: Limit Superior And Limit Inferior, Bounded Function, Joule

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A collection of problems for the final exam. Linde april 28, 2016: let f be the function de ned by f (x) = 1 + |x + 1| x r . Determine the maximal value of f and nd intervals where f is decreasing or increasing: let f be a function from a set x into r such that inf x x f (x) > 0. Let g(x) := 1/f (x), x x. Show that g is bounded above and that. 1 g(x) = sup x x inf x x f (x: suppose f : r (0, ) is continuous with lim x f (x) = 0 and lim x f (x) = 0 . Show that lim n xn = sup n 1 xn : determine inf x r. 1 + (1 x)2 and sup x r. 1 + (1 x)2 : find the following limits and prove (with , and/or ) the results. x2.

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