limits_of_reals page 6.pdf - 6 Proof Let L = lim an If we set = 1 then by the definition of limit there is some N > 0 such that |an n!1 L| < 1 for all n

# limits_of_reals page 6.pdf - 6 Proof Let L = lim an If we...

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6 Exercises for Section 2 A.Prove parts (2) and (3) of Theorem 2.1.B.Compute the following limits.(a)limn!13n5-n4+ 1757n5+ 100N3-32n(b)limn!12100+5ne4n-10(c)limn!12nn!+2 arctannlognC.Iflimn!1an=L >0, prove thatlimn!1pan=pL. Be sure to discuss the issue of whenpanmakessense.Hint: Express|pan-pL|in terms of|an-L|.D.Let (an)1n=1and (bn)1n=1be two sequences of real numbers such that|an-bn|<1n. Suppose thatL= limn!1anexists. Show that (bn)1n=1converges toLalso.E.Findlimn!1log(2 + 3n)2n.Hint: log(2 + 3n) = log 3n+ log2+3n3nF.(a) Letxn=npn-1. Use the fact that (1 +xn)n=nto show thatx2n2/n.