Prove using the definition of a limit that limnrarrinfin n

Question

Prove, using the definition of a limit, that limn→∞ n n2 + 1 = 0. 2. Prove, using the definition of a limit, that limn→∞ n 2 + 1 n2 + 3 = 1. 3. Let c be a fixed positive number and an = 1 1 + nc where c > 0. Use the definition of a limit to prove that ?an? converges. 4. Prove that the sequence ? √ n? diverges.