Notation xv Given two functions f and g defined on [a, ∞) (where a ≥ 0), we shall write f(x) g(x) if there exist two constants M 0 and x0 such that |f(x)| M|g(x)| for all x ≥ x0. Thus, we have x √ x, 2 sin x 1, √ x log x, 1 + sin x x 1, ex x4, xex ex. On the other hand, given two functions f and g defined on [a, ∞) (where a ≥ 0), we write f(x) ∼ g(x) to mean that lim x→∞ f(x) g(x) = 1. Thus, as x → ∞, sin1/x 1/x ∼ 1, x2 + x ∼ x2. Finally, we write that f(x) g(x) if we simultaneously have f(x) g(x) and g(x) f(x). Observe that f(x) g(x) if and only if 0 lim inf x→∞ f(x) g(x) ≤ lim sup x→∞ f(x) g(x) ∞.
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