# Homework Help: To prove that these two functions meet only once

1. Mar 11, 2010

### willy0625

1. The problem statement, all variables and given/known data

How would you go about proving that the functions

$$F(x)=\frac{2}{\pi}\arctan\Big(\frac{x}{a}\Big), x \geq 0$$

with $$a > 0$$

and

$$G(x)=1-\exp(-\lambda x),x\geq0$$

with $$\lambda>0$$

meet only at one point for some $$x > 0$$

3. The attempt at a solution

At $$x=0$$, F and G takes the values 0s and both of the functions tend to 1 when x gets really large I think they must meet at least once. But still not sure about finding the actual point(s) since i do not think i can solve the equation
$$F=G$$ analytically.

Last edited: Mar 11, 2010
2. Mar 11, 2010

### Sweet_GirL

as x becomes large, the two function tend to 1.
And clearly they are increasing functions.

so what do you suggest?

3. Mar 11, 2010

### Tinyboss

I don't believe the problem is asking you to find x, rather to show it exists and is unique.

4. Mar 11, 2010

### willy0625

Yeah showing the uniqueness of that particular point is enough.

Last edited: Mar 11, 2010
5. Mar 11, 2010

### Tinyboss

The fact that your functions have the same limit does not mean they ever meet. Take for instance 1/x and 1/(x+1), for x>0. Clearly they both tend to zero, and clearly they are never equal.

I don't see right away how to do existence or uniqueness. By continuity if you can find a point where F<G then you have existence, and then maybe you could look at the derivative of F-G to show uniqueness. Or maybe there's something prettier. What topics has your classed covered recently?

6. Mar 11, 2010

### willy0625

This question poped up somewhere in my research.
I can use any results in calculus and analysis.

Note I have made some correction to the F function.

Last edited: Mar 12, 2010