test1-0607 - Department of Mathematics University of...

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Department of Mathematics, University of Toronto Term Test 1 – November 15, 2006 MAT 137Y, Calculus! Time Alloted: 1 hour 50 minutes 1. Evaluate the following limits. Do not use L’Hˆopital’s Rule to evaluate the limit. (7%) (i) lim x 0 ( x - 1 ) 2 - 1 x 2 + 6 x . (7%) (ii) lim t 0 sin 2 ( 5 t ) 3 t 2 . (7%) (iii) lim x 4 + ( 4 - x ) | 3 x - 14 | | 4 - x | . (7%) (iii) lim x 0 3 - 9 - x 2 x 2 . 2. (7%) (i) Solve the inequality x 2 - 3 x x 4 - 1 0. Express your answer as a union of intervals. (ii) Suppose sin x = 3 4 and π 2 x π . Find the exact value of each of the following expres- sions. (6%) (a) tan x . (4%) (b) cos2 x . 3. (5%) (a) Give the precise ε , δ definition of the following statement: lim x a f ( x ) = L . (12%) (b) Prove that lim x 3 x 2 + 1 1 - x = - 5 directly using the precise definition of limit. 4. Consider the sequence of numbers x 1 = 1 , x 2 = q 1 + 1 , x 3 = r 1 + q 1 + 1 , x 4 = s 1 + r 1 + q 1 + 1 ,..., so x n containes n nested radicals and exactly n ones. (3%) (a) Express x n in terms of x n - 1 . (9%) (b) Prove for all positive integers n 2 that x n is irrational. 1
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5. For each of the following statements determine whether the statement is true or false (by putting a checkmark next to the word “True” or “False”) and then justifying your answer
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