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Question

Let m and n be two positive integers greater than 1. If
limα0ecos(αn)eαm=e2, then the value of mn is

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Solution

limα0ecos(αn)eαm=elimα0ecos(αn)11αm=elimα0ecos(αn)11cos(αn)1×cos(αn)1αm=elimα0cos(αn)1αm=elimα02sin2αn2αm=elimα02sin2αn24(αn2)2×α2nαm=e2α2nm=e22nm=0mn=2

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