MthT 430 Notes Chapter 9 Limits and Order

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" 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. An expression (function) f(x) is little o of x as x ® 0, written f(x) = o(x) [as x ® 0], if

lim x ® 0  f(x) |x| = 0.

If we are not worried about the particular details of f(x), we write f(x) = o(x) [as x ® 0].

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_{x ® 0} y(x) = 0, we write y(x) = o(1) with the precise meaning that

lim x® 0  y(x) 1 = 0.

Definition. Let q(x) be nonzero for x near and not equal 0. Then a function f(x) is little o of q(x), written f(x)=o(q(x)), if

lim x ® 0  f(x) |q(x)| = 0.

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

f(x) |q(x)|

is bounded as x ® 0.

N.B. We are assuming that, for x small enough and ¹ 0, both f(x) and q(x) are defined and q(x) ¹ 0.

We could also propose an e-d definition of little o(q):

Definition. A function f = o(q) as x ® 0 means For every e > 0, there is a d > 0 such that if 0 < |x| < d, then |f(x)| < e|q(x)|

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

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

as Dx® 0.

A real valued function of a real variable x is differentiable at x with derivative f^¢(x) if

f(x+Dx) = f(x) +f¢(x) Dx+ o(Dx)

as Dx® 0.

Local boundedness of a function can be expressed as f(x+Dx)=O(1) as Dx® 0.

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).

Proof of the Chain Rule

The Chain Rule. Let g(z) be differentiable at z, and let f(w) be differentiable at w = g(z). Then h(z) = f(g(z)) is differentiable at z and

h¢(z) = d dz f(g(z)) = f¢(g(z))·g¢(z).

Proof: We show that

h(z + Dz) = f(g(z + Dz)) = f(g(z)) + f¢(g(z)) ·g¢(z)·Dz+ o(Dz).

as Dz® 0.

Let

Dg(z) = g(z + Dz) - g(z) = g¢(z) Dz+ o(Dz).

We are assuming that

g(z+Dz) = g(z) +g¢(z) ·Dz+ o(Dz), f(g(z)+Dg(z)) = f(g(z)) +f¢(g(z)) ·Dg(z) +o(Dg(z)).

Since

Dg(z) = g¢(z) Dz+o(Dz) = O(Dz), o(Dg(z)) = o(O(Dz)) = o(Dz),

we have

f(g(z + Dz)) = f(g(z)) +f¢(g(z)) ·g¢(z) ·Dz+ o(Dz).

Remarks

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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On 06 Nov 2007, 21:32.