Example:<\/strong> Evaluate limn\u2192\u221e<\/sub>n^(1\/n)<\/p>\n Solution:<\/p>\n First, imagine<\/p>\n y=limn\u2192\u221e<\/sub>n^(1\/n)<\/p>\n Next, we take the logarithm of both sides:<\/p>\n lny=lnlimn\u2192\u221e<\/sub>n^(1\/n)<\/p>\n Now, because both lny and n^(1\/n) are continuous n\u2192\u221e, we can change their order on the right side:<\/p>\n lny=limn\u2192\u221e<\/sub>lnn^(1\/n)<\/p>\n Using the log rule about exponent-to-multiple gives<\/p>\n lny=limn\u2192\u221e<\/sub>(1\/n)lnn<\/p>\n or<\/p>\n lny=limn\u2192\u221e<\/sub>lnn\/n<\/p>\n This limit has the \u221e\/\u221e form, which means, by l’H\u00f4pital’s Rule, we can take the derivative of the numerator and denominator separately, then take the limit of that result:<\/p>\n lny=limn\u2192\u221e<\/sub>(1\/n)\/1=limn\u2192\u221e<\/sub>(1\/n)<\/p>\n So,<\/p>\n lny = 0<\/p>\n We take the exponential of both sides:<\/p>\n e^(lny) = e^0<\/p>\n to arrive at<\/p>\n y=1<\/p>\n Recalling that y=limn\u2192\u221e<\/sub>(1\/n), we realize that<\/p>\n limn\u2192\u221e<\/sub>n^(1\/n)=1<\/p>\n HTH:)<\/p>\n Source:<\/p>\n Larson, Roland E. and Robert P. Hostetler. Calculus<\/u>. Toronto:
\u00a0\u00a0 D.C. Heath and Company, Ltd., 1986.<\/p>\n