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# smidterm1 - lim n →∞ a n = L 1 2 CALCULUS 153 SAMPLE...

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CALCULUS 153: SAMPLE MIDTERM 1 Problem 1 (16 points) . Determine the least upper bound and greatest lower bound of the following sets, or state that they do not exist. You do not need to justify your answer (4 points each). (1) ( -∞ , 3] [4 , 7) , (2) { x : 2 < ln( x + 1) < 4 } , (3) { a n } where a n = 2 1 /n , (4) { x Q : e < x < π } . Problem 2 (20 points) . For each of the following sequences, determine whether the sequence converges or diverges. If it converges, ﬁnd its limit. Show your work. (5 points each). (1) a n = 2 n + ( - 1) n n ! ; (2) b n = ± 1 + x n ² 2 n ; (3) c n = sin( n 2 + ln n ) n 1 /n n + 1 . (4) d n = x n n +1 where x > 0 . Problem 3 (20 points) . Find the following limits. Show your work (5 points each). (1) lim x 0 = e x - e - x sin x , (2) lim x 0 + x sin x , (3) lim x 1 1 x - 1 - x ln x , (4) lim x 0 1 + x - e x x ( e x - 1) . Problem 4 (10 points) . State the ± K deﬁnition of

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Unformatted text preview: lim n →∞ a n = L . 1 2 CALCULUS 153: SAMPLE MIDTERM 1 Problem 5 (14 points) . Prove that if T is a subset of S , then sup T ≤ sup S . Problem 6 (20 points) . For each of the following give an example, or state that no example exists (you do not need to justify your answer). (1) A set S whose least upper bound is not an element of S . (2) A set of negative numbers whose least upper bound is strictly positive. (3) A set of rational numbers whose least upper bound is irrational. (4) A set of irrational numbers whose least upper bound is rational. (5) A sequence that’s bounded but divergent....
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smidterm1 - lim n →∞ a n = L 1 2 CALCULUS 153 SAMPLE...

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