Using the method of Example i, prove that the following limits exist.a) 2 − 1/n → 2 as n→∞...

Using the method of Example i, prove that the following limits exist.

a) 2 − 1/n → 2 as n→∞.

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b) as n→∞.

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c) 3(1 + 1/n) → 3 as n→∞.

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d) (2n2 + 1)/(3n2) → 2/3 as n→∞.

EXAMPLE

i) Prove that 1/n → 0 as n→∞.

Proof. i) Let ε > 0. Use the Archimedean Principle to choose N ∈ N such that N > 1/ε. By taking the reciprocal of this inequality, we see that n ≥ N implies 1/n ≤ 1/N < ε. Since 1/n are all positive, it follows that |1/n| < ε for all n ≥ N.

Strategy for ii): By definition, we must show that

is small for large n. The numerator of this last fraction will be small for large n since xn → 2, as n→∞. What about the denominator? Since xn → 2, xn will be greater than 1 for large n, so 2xn will be greater than 2 for large n. Since we made n large twice, we will make two restrictions to determine the N that corresponds to ε in Definition 2.1. Let’s try to write all this down carefully to be sure that it works out.