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)

∞.