Evaluate Limit of f(x) as x→∞

[longrob]

Homework Statement

Evaluate the limit of the following as x approaches infinity

$$\frac{e^{x}-1}{1-2e^{x}+2e^{2x}}$$

Homework Equations

The Attempt at a Solution

$$\frac{e^{-x}-e^{-2x}}{e^{-2x}-2e^{-x}+2}$$
which gives 0/2=0 as x approaches infinity, but apparently this is wrong.

 

[Dick]

Why do you think it's wrong? I think it's right.

 

[longrob]

Because I am told the answer is $$\frac{1}{2}e^{-x}$$

 

[Dick]

The limit of (1/2)e^(-x) as x->infinity is 0. Are they actually asking for the asymptotic behavior or the limiting behavior? Not the limit?

 

[longrob]

The question doesn't explicitly ask for the limit, I just assumed that's what I had to do. This is an applied maths course, not analysis. The question is asking for the long term behaviour of the solution to a differential equation...

 

[longrob]

in case I'm not being clear, the expression I posted initially is the general solution to a DE problem that I obtained (so it might be wrong !) and the question is to show that the long term behaviour as x->infinitty is that x approaches (1/2)e^(-x)

 

[Dick]

That would make sense. But I'm not sure what they mean by show. Casually you would just pick the dominant term in the numerator (e^x) (dominant meaning the ratio of any other term in the numerator and that one goes to zero) and the dominant term in the denominator (2e^(2x) and divide them. If they want you be more formal they might want you to show limit f(x)/g(x) goes to 1, where f(x) is your original expression and g(x)=(1/2)e^(-x).

 

[longrob]

Thank you very much.

 

[longrob]

Dick, Your final remark about the more formal approach requires me to know the answer, ie g(x) already, which was obtained by the "casual" approach. So how do you find g(x) more formally than using the casual approach ? Thanks a lot !

 

[Dick]

I don't think there is any 'formal' way to find g(x). You just throw out parts of the function that are less important. The only question was do you have to PROVE that what you've kept of the function is the important part? In that case showing limit f(x)/g(x)->1 would be one way.

 

[longrob]

Got it. Thanks again.