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Asymptotics Sheet 1 1. For 1, obtain two-term expansions for the solutions of the following equations (a) (x − 1)(x − 2)(x − 3) + = 0. (b) x3 + x2 − = 0. (c) x3 + x2 + 2x + 1 = 0. √ (d) 2 sin(x + π/4) − 1 − x − 12 x2 = − 61 , (only the solution near x = 0). (e) x3 + x2 + (2 + )x + 1 = 0. (f) x3 + x2 + (2 − )x + 1 = 0. (g) Bonus(+20%): 2 x3 + x2 − 6x + 1 = 0. 2. Find the rescaling for the roots of 2 x3 + x2 + 2x + = 0 and thence find two terms in the approximation for each root. 3. Verify the following statements (a) sin x1/3 = O x1/3 , (b) cos x = O(1), x → ∞. (c) sin x = O(x cos x), x → 0+. x → 0. (d) log log x1 = o(log(x)), x → 0+. 4. Explain why the sequence {φn (x) = x−n cos(nx)}, n = 0, 1, . . ., is not and asymptotic sequence as x → ∞. 5. Prove that 1 n=1 z n P is an asymptotic expansion of 1 z−1 as z → ∞. 6. Show, by finding a counterexample, that f (x) ∼ g(x) as x → x0 doe not necessarily imply that ef (x) ∼ eg(x) x → x0 .