MAT137Y1 Midterm: 2004 Test 1 solution
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MAT137Y1 Full Course Notes
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Mat 137y 2004-2005 solutions to term test 1: evaluate the following limits, or show that the limit does not exist. (10%) (i) lim x 0. = lim x 0 (cid:19) (1 + x) (1 x) = 1. (10%) (ii) lim x 0 (tan x)(tan2x) x tan3x (tan x)(tan2x) lim x 0 x tan3x. By de nition, |4 x| = x2 x 12 lim x 4+ x 4, |4 x| x 4, x < 4. Taking cases on x, we have x2 x 12 x 4 (x 4)(x + 3) , it follows the limit does not exist. 2. (8%) (i) solve the inequality x2 9. Moving all terms to one side, we solve x2 9. 1 < 0 x2 9 (3x + 1) 3x + 1 x 5 x + 2. 3 < x < 5 x > 5 (7%) (ii) find the least upper bound of the set s = {1 1.
