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 .