# A3_2011 - ∞ EXERCISES You do not need to submit these...

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University of Toronto at Scarborough MAT A37H Summer 2011 Assignment # 3 You are expected to work on this assignment prior to your tutorial during the week of May 30th. You may ask questions about this assignment in that tutorial. At the beginning of your TUTORIAL during the week of June 6th you need to submit the following homework problems. STUDY: Chapter 5 (pgs 98-105) HOMEWORK PROBLEMS: 1. Prove that lim x 3 - log 2 (3 - x ) = -∞ . 2. Prove that: lim x →∞ x - sin( x + 1) 3 x + 9 = 0 . 3. Prove that: lim x →∞ 4 x 2 - 7 2 x 3 - 5 = 0. 4. Let g,f : R R be functions and let a,‘,m R . If lim x →∞ g ( x ) = and lim x →∞ f ( x ) = m , then give a complete N - ± proof that lim x →∞ ( g ( x ) - af ( x )) = - am . 5. Suppose that lim x a f ( x ) = and lim x a g ( x ) = π , where a R . Prove lim x a f ( x ) g ( x ) =

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Unformatted text preview: ∞ . EXERCISES: You do not need to submit these questions but you should make sure that you are able to answer them; 1. Spivak # 5.8, 5.9, 5.10(a), 5.12, 5.13, 5.16, 5.18, 5.33(i)-(iii), 5.38(b), 5.39(v)-(vi). 2. Give a precise deﬁnition for each of the following limits: (a) lim x → a f ( x ) =-∞ , where a ∈ R . (b) lim x → a + f ( x ) =-∞ , where a ∈ R . (c) lim x →∞ f ( x ) = ∞ . (d) lim x →-∞ f ( x ) = ‘ , where ‘ ∈ R . 3. Let a, L ∈ R . Suppose that lim x → a f ( x ) = L. Provide a complete and accurate δ-± proof that lim x → a ( f ( x )) 2 = L 2 . 4. Show that: lim x → 3 1 ( x-3) 2 = ∞ . (This is Spivak # 5.37(a)). 2...
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A3_2011 - ∞ EXERCISES You do not need to submit these...

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