Limits and Order

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

f¢(x)= lim Dx ® 0  f(x+Dx) - f(x) Dx ,

which means that the difference

f(x+Dx) - f(x) Dx - f¢(x)

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

Multiplying the remainder by Dx, we obtain that

f(x+Dx) - f(x)-f¢(x) Dx= small·Dx,

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

lim Dx ® 0  RHS |Dx| =0.

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

Definition. As Dx® 0, an expression f(Dx) is little o of Dx, written o(Dx), if

lim Dx ® 0  f(Dx) |Dx| = 0.

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

f(x+Dx) = f(x) +f¢(x)·Dx+ o(Dx).

In a similar way, if lim_{Dx ® 0} y(Dx) = 0, we write y(Dx) = o(1) with the precise meaning that

lim Dx ® 0  y(Dx) 1 = 0.

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

lim Dx ® 0  f(Dx) |q(Dx)| = 0.

Then a function f(x) is big O of q(Dx), written f(Dx)=O(q(Dx)), if

f(Dx) |q(Dx)|

is bounded as Dx ® 0.

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

f(x+Dx) = f(x) + o(1),

and local boundedness of a function can be expressed as f(x+Dx)=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(Dx) = o(Dx); i.e., the product of a bounded function and a function which is o(Dx) is o(Dx). Similarly o(Dx) ±o(Dx) = o(Dx).

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 3.76.
On 26 Apr 2007, 15:49.