MATH 1210 Study Guide - Midterm Guide: Mean Value Theorem, Partial Fraction Decomposition, Indeterminate Form
23 Mar 2017
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When limits are in an indeterminate form, we cannot just plug in to solve the limit. , 0 , , 00, 0, 1 can be made easier to solve using cauchy"s mean value theorem and l"h opital"s rule. Begin with the conditions for cauchy"s mean value theorem; let f and g be continuous and differentiable functions on [a, b], and let g (x) = 0 x (a, b). This implies there is at least on number c such that f (c) g (c) f (b) f (a) g(b) g(a) The proof for l"h opital"s rule requires an interval i with the same properties as cacuhy"s mean value theorem. For x i, we have two cases: x > a (x a+) and x < a (x a ). Case 1: x > a, we"ll say there is a subinterval [a, x]. We have f (c) g (c) f (x) f (a) g(x) g(a)
