IMPORTANT LIMITSDAVID BEN MCREYNOLDS1. Important Limits(1) If x > 0, thenx1/n−→ 1 as n −→ ∞(2) If |x| < 1, thenxn−→ 0 as n −→ ∞(3) For each α > 0,1nα−→ 0 as n −→ ∞(4) For each real x,xnn!−→ 0 as n −→ ∞(5)ln nn−→ 0 as n −→ ∞(6)n1/n−→ 1 as n −→ ∞(7) 1 +xn!n−→ exas n −→ ∞2. Whose BiggerHere is a list of whose bigger than who.ln n < n < n2< · · · < n100< · · · < 2n< 3n< . . . 100n< . . .< n! < n!2n< nn< 3nn! < n2n< . . .There is no biggest, so our lis t goes on and on and on, like theEnergizer Bunny.Date: January 24, 2002.12 DAVID BEN MCREYNOLDSIf a sequence ancomes before bnin the list, thenlimn−→∞anbn= 0As an example, n100comes before 2n, solimn−→∞n1002n= 0Likewise, if ancomes before bn, thenanbn−→ ∞ as n −→ ∞This is not a way to do any problems. This is to help you know theanswer before you actually do the problem. You still need to use thetechniques of the book to show the limit.University of Texas at Austin, Department of MathematicsE-mail address:
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