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# A2_2011 - 2 Spivak 2.8 2.12 2.13 3.3(i(iii 3.16 3.19(i...

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University of Toronto at Scarborough Department of Computer & Mathematical Sciences MAT A37H Summer 2011 Assignment # 2 You are expected to work on this assignment prior to your tutorial during the week of May 23rd. You may ask questions about this assignment in that tutorial. At the beginning of your TUTORIAL during the week of May 30th you need to submit the following homework problems. STUDY: Chapter 2 (pg 25-26), Chapters 3-4. Chapter 5 (excluding pgs 100-107). HOMEWORK PROBLEMS: 1. Prove that 2 + 12 is irrational. 2. Spivak # 3.3(iv). 3. Prove that lim x 2 x 4 - 2 x 3 + x + 3 = 5 . 4. Prove that lim x 3 x 2 + 1 1 - x = - 5 . 5. Consider the function f ( x ) = 2 x if x Q - 2 x if x / Q Show that lim x 0 f ( x ) = 0 . EXERCISES: You do not need to submit these questions but you should make sure that you are able to answer them;

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1. Let P ( n ) denote ”7 n - 2 n = 5 x ” for some x N . Prove P ( n ) holds n N by using the Principle of Well Ordering (PWO).
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Unformatted text preview: 2. Spivak # 2.8, 2.12, 2.13, 3.3(i)-(iii), 3.16, 3.19(i), 3.21(a)(b). 3. Let A = { 1 , 3 , 4 , 5 , 8 } and B = { 2 , 3 , 4 , 5 } . The set { (1 , 4) , (5 , 2) , (4 , 5) , (3 , 3) , (8 , 4) } deﬁnes a function from A into B . T or F 4. Find the domain of the function f ( x ) = q x-1 x +1 + 1 log 4 (4-2 x ) . 5. Prove that: (a) lim x → a √ x = √ a where a ∈ R ,a > . (b) lim x → 1 2 x 2-3 x + 1 x-1 = 1 . (c) lim x → 2 1 ( x-3) 2 = 1 . (d) lim x → 3 x 3-2 x 2 + x-1 = 11 . (e) lim x → x cos ± 6 x ² = 0 . (f) lim x → 1 ( x 2-1) sin ± 1 x-1 ² = 0 . 6. Let f : R → R be a function and let a ∈ R , ‘ ∈ R . Give a precise deﬁnition of f does not approach ‘ at a . ie. lim x → a f ( x ) 6 = ‘. Make sure to ”simplify”/negate your deﬁnition correctly. 2...
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