Can anybody check my proof?


by Artusartos
Tags: check, proof
Artusartos
Artusartos is offline
#1
Nov8-12, 03:31 AM
P: 245
Let f be differentiable on some interval (c, infinity) and suppose that [tex] lim_{x \rightarrow \infty} [f(x) + f'(x)] = L [/tex], hwere L is finite. Prove that [tex]lim_{x \rightarrow \infty} f(x) = L[/tex] and [tex] lim_{x \rightarrow \infty} f'(x) = 0 [/tex]. Hint: [tex] f(x) = \frac{f(x)e^x}{e^x}[/tex]

My answer:

For [tex]f(x) = \frac{f(x)e^x}{e^x}[/tex], Let h(x)=f(x)e^x and let g(x)=e^x. So we have [tex]f(x) = \frac{h(x)}{g(x)}[/tex] Since we know that the sum of the limits of f(x) and f'(x) is finite, we know that each limit must also be finite. Therefore, h(x) eventually be less than g(x). So for large x, h(x) =< g(x) in order for h(x)/g(x) to converge. Since h(x)=f(x)e^x and g(x)=e^x, f(x) =< 1.

This can happen in two ways. Either h(x) is [itex]ke^x[/itex] where k is a constant such that |k|=< 1...

If this is the case, then the derivative of f(x) converges to zero, while f(x) itself converges to some number L.

...or f(x) (in f(x)e^x) is a decreasing function that converges to a number that is less than or equal to 1.

In this case, both f(x) and f'(x) converge to zero.

Do you think my answer is correct?
Phys.Org News Partner Science news on Phys.org
Better thermal-imaging lens from waste sulfur
Hackathon team's GoogolPlex gives Siri extra powers
Bright points in Sun's atmosphere mark patterns deep in its interior
clamtrox
clamtrox is offline
#2
Nov8-12, 05:33 AM
P: 937
Quote Quote by Artusartos View Post
Since we know that the sum of the limits of f(x) and f'(x) is finite, we know that each limit must also be finite.
It seems to me that this is exactly what you needed to show in the first place. Can you show this is true?

Or if you want an easier approach, that hint suggests to me that you might want to use L'Hopital.
hedipaldi
hedipaldi is offline
#3
Nov8-12, 07:11 AM
P: 206
The limit of a sum may be finite even if neither limit is finite.Take x and -x.


Register to reply

Related Discussions
Need someone to check my proof... Calculus & Beyond Homework 2
Please Check proof. Calculus & Beyond Homework 8
Lebesgue measurability proof - check my proof? Calculus & Beyond Homework 2
Can someone check my proof? Set Theory, Logic, Probability, Statistics 5
Can you check my proof? Differential Geometry 5