test1-04 - a n = F n F n 1 4 Consider the statement lim x...

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Department of Mathematics, University of Toronto Term Test 1 – November 5, 2004 MAT 137Y, Calculus! Time Alloted: 1 hour 50 minutes 1. Evaluate the following limits. (No proofs are required.) (10%) (i) lim x 0 1 + x - 1 - x x . (10%) (ii) lim x 0 ( tan x )( tan2 x ) x tan3 x . (10%) (iii) lim x 4 | 4 - x | x 2 - x - 12 . 2. (8%) (i) Solve the inequality x 2 - 9 3 x + 1 < 1. (7%) (ii) Find the least upper bound of the set S = { 1 - 1 n 2 : n = 1 , 2 ,... } and justify your answer. 3. Recall the Fibonacci Sequence { F n } is defined recursively as follows: F 1 = 1, F 2 = 1, and F n + 2 = F n + 1 + F n for all integers n 1. (7%) (a) Prove for all integers n 1 that F n > 0. (8%) (b) Let { a n } be the sequence defined recursively by a 1 = 1, a k + 1 = 1 1 + a k . Prove for all integers n 1 that
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Unformatted text preview: a n = F n F n + 1 . 4. Consider the statement: lim x → a f ( x ) = L . (5%) (a) Give the formal ε , δ definition of the statement above. (10%) (b) Give a formal ε , δ proof that lim x → 3 x 2 + 7 x + 1 = 4. 5. (5%) (a) State the Intermediate Value Theorem. (10%) (b) Prove that there exists a non-zero solution to the equation sin x = x 2 . (10%) 6. Suppose we are given a function f such that for all values a , b ∈ R , | f ( b )-f ( a ) | ≤ | b-a | . Prove that f must be continuous for all x . 1...
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This note was uploaded on 04/09/2010 for the course MAT 137 taught by Professor Uppal during the Spring '08 term at University of Toronto.

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