Optimizing f(p,q) with Close Proximity of p and q in (0,1): A Formal Analysis

[jimholt]

Homework Statement

I have the function f(p,q)=p(1-p)/[q(1-q)] where p and q are in (0,1). I want to say that if p is close to q, f(p,q) is 'close' to 1. What is a formal way of saying how close to p q should be?

The Attempt at a Solution

Basically I want to say $$f(p,q)= 1 + \epsilon(p,q)$$ where $$\epsilon(p,q)$$ is small if some condition "X" is true. Is there an obvious way of saying what "X" is? Maybe I can substitute q=p+u... but then what? Taylor series or L'hopital's or something? My calc is pretty rusty, so I'd appreciate any reminders...

 

[Pere Callahan]

Substituting p = q+u sounds like a good idea. Then use
$$(q+u)(1-q-u) = q-q^2-2qu+u-u^2 = q(1-q)-2qu+u-u^2$$

 

[Dickfore]

If you define a function

$$g(x) = x (1 - x)$$

can you translate the question you asked in terms of it?

 

[Raskolnikov]

Why not just write that the limit[f(p,q)] = 1 as p $$\rightarrow$$ q ?

 

[paulfr]

Why not just write that the limit[f(p,q)] = 1 as p $$\rightarrow$$ q ?

I agree.
The problem statement is nearly the very definition of a Limit.
You just did not use the letters epsilon and delta x.

 

[HallsofIvy]

$$\frac{p(1-p)}{q(1- q)}= \left(\frac{p}{q}\right)\left(\frac{1-p}{1-q}\right)$$
Does that help?