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)
∞.
