Limits and Order

For functions of a real variable, the derivative is defined as

f′(x)= lim ∆x → 0  f(x+∆x) − f(x) ∆x ,

which means that the difference

f(x+∆x) − f(x) ∆x − f′(x)

is small if ∆x is small and not 0 (for which the quotient is not obviously defined).

Multiplying the remainder by ∆x, we obtain that

f(x+∆x) − f(x)−f′(x) ∆x= small·∆x,

with the right hand side, (RHS), of the equation is "much smaller than ∆x", as ∆x→ 0, in the precise sense

lim ∆x → 0  RHS |∆x| =0.

Another formal advantage is that the equation is also defined and true for ∆x = 0.

Definition. As ∆x→ 0, an expression ϕ(∆x) is little o of ∆x, written o(∆x), if

lim ∆x → 0  ϕ(∆x) |∆x| = 0.

If we are not worried about the particular details of ϕ(x), we write ϕ(x) = o(∆x). With this convention, the definition of differentiability and the derivative takes the convenient form

f(x+∆x) = f(x) +f′(x)·∆x+ o(∆x).

In a similar way, if lim_{∆x → 0} ψ(∆x) = 0, we write ψ(∆x) = o(1) with the precise meaning that

lim ∆x → 0  ψ(∆x) 1 = 0.

Definition. Let q(∆x) be nonzero for ∆x near 0. Then a function ϕ(∆x) is little o of q(∆x), written ϕ(∆x)=o(q(∆x)), if

lim ∆x → 0  ϕ(∆x) |q(∆x)| = 0.

Then a function ϕ(x) is big O of q(∆x), written ϕ(∆x)=O(q(∆x)), if

ϕ(∆x) |q(∆x)|

is bounded as ∆x → 0.

With this convention, continuity of a function f(x) can be expressed by

f(x+∆x) = f(x) + o(1),

and local boundedness of a function can be expressed as f(x+∆x)=O(1).

There is a formal calculus for handling sums and products for functions which are little o or big O of one (or several) q. Verify that O(1) ·o(∆x) = o(∆x); i.e., the product of a bounded function and a function which is o(∆x) is o(∆x). Similarly o(∆x) ±o(∆x) = o(∆x).

The concepts little o and big O are also useful as the argument x → ∞. For example we write x²=o(e^x) as x→ ∞ with the precise meaning

lim x→∞  x2 ex =0.

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File translated from T_EX by T_TH, version 4.03.
On 16 Nov 2013, 20:43.