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MATH 101 HOMEWORK 2 Due on Wednesday Sept. 22. For full credit, show all work. Calculators are not allowed. 1. (6 marks) Evaluate the limits: n + 2n2 − n3 , n→∞ 1 + 2n3 (a) lim 1 − cos(2/n) . n→∞ 1 − cos(3/n) (b) lim 2. (6 marks) Evaluate the sums: (a) 92 + 122 + 152 + 182 + . . . + (3n − 3)2 + (3n)2 , n X i X ( 2i+j ). (b) i=1 j=1 R5 3. (4 marks) Write the upper and lower Riemann sums approximating 2 x−3 dx, corresponding to the partition of [2, 5] into n intervals of equal length. (Do not try to evaluate the sums.) 4. (4 marks) Identify the limit lim n→∞ n X 3n − 2i i=1 n2 as an area of a planar region, and use this to evaluate it. 1