Practice Assign3 - 137 - F20 SOLUTIONS.pdf - MATH 137...

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MATH 137 PRACTICE ASSIGNMENT 3 SOLUTIONS 1. Assuming lim n →∞ a n = lim n →∞ b n = 0 where a n 0 and b n 0, determine lim n →∞ ( a n sin( n ) + b n cos( n )) .
2. Let’s examine how absolute values and limits interact. (a) The statement If lim n →∞ | a n | = | L | then lim n →∞ a n = L is false in general. Provide a counter-example. (b) The statement If lim n →∞ a n = L then lim n →∞ | a n | = | L | is true. Show this using the definition of limits. Hint: || a | - | b || ≤ | a - b | (Not required: Can you show that the Hint is true?) (c) Is the statement If lim n →∞ | a n | = 0 then lim n →∞ a n = 0 true? If so, argue why, if not, provide a counterexample. Solution

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