limit c test - Math 101 Fall 2010 The Improved Limit Comparison Test for Improper Integrals Section 8.7 Theorem 3 The Limit Comparison Test Let f and g

limit c test - Math 101 Fall 2010 The Improved Limit...

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Math 101 – Fall 2010 The Improved Limit Comparison Test for Improper Integrals Section 8.7 Theorem 3* – The Limit Comparison Test:Letfandgbe nonnegativefunctions on [a,) , and supposeL= limx→∞f(x)g(x).Then:a.If 0< L <, then:af(x)dxconverges⇐⇒ag(x)dxconvergesaf(x)dxdiverges⇐⇒ag(x)dxdivergesb.IfL= 0 , then:ag(x)dxconverges =af(x)dxconvergesaf(x)dxdiverges =ag(x)dxdivergesc.IfL=, then:af(x)dxconverges =ag(x)dxconvergesag(x)dxdiverges =af(x)dxdivergesRemark:Similar tests are valid for the other types of basic improper integrals.Examples:Determine whether each of the following improper integrals converges or diverges. 2L’H=limx→∞61/8ex/2= 0.We also havee-x/2dx= limc→∞ce-x/2dx= limc→∞[-2e-x/2]c= limc→∞(-2e-c/2+ 2) = 2.SinceL= 0 ande-x/2dxconverges, we conclude thatx3e-xdxconverges by the LimitComparison Test.2.πsin2xx2dx:We haveL= limx→∞sin2x/x21/x3/2= limx→∞sin2xx1/2= 0by the Sandwich Theorem. We also know that thep-integralπdxx3/2converges asp= 3/2>1 .Thereforeπsin2xx2dxconverges by the Limit Comparison Test.