Applying the MVT to Show f(x)/x Goes to b When x Goes to Infinity

[blinktx411]

Excuse the typing please, as I am posting from my phone.
Let f have domain [0,infty) and range in R. Suppose as x goes to infinity, f'(x) goes to a constant b. I wish to show that f(x)/x goes to b as x goes to infinity.

I have tried numerous applications of the MVT to solve this and cannot come up with f(x)/x. What I can show is for every h>0, (f(x+h)-f(x))/h goes to b as x goes to infinity. Does anyone see or can give me a hint on how to apply the MVT here? Thanks.

 

[snipez90]

Hmm, I don't see immediately how the MVT argument would work. I've certainly used MVT before to prove something about g(x) = f(x)/x by applying MVT to g over the interval [0,x], but this usually requires the assumption that f(0) = 0 (so we have the required quotient).

But actually l'hopital's rule can be applied here, I think. One thing that I didn't know before I studied analysis was that for the infinity/infinity indeterminate form case, the hypothesis that the function in the numerator has to approach infinity is superfluous. In other words, if g(x) = f(x)/h(x), and we know that f'(x)/h'(x) tends to a limit, then the only additional hypothesis needed to apply l'hopital is that h(x) approaches infinity (or negative infinity). In particular, we don't even need to know anything about the limit of f(x).

The proof of this is found in Rudin's Principles of Mathematical Analysis, and it probably uses the Cauchy MVT. The argument can be adapted as this problem appears to be a special case of the theorem.

 

[blinktx411]

Thank you! The problem is from bartle, 27s if you're interested. The context suggests I should be able to prove this from MVT directly, but so far no luck. I'm going to take a closer look at your suggestion, and thanks!

 

[snipez90]

OK, I found the problem on google books. Apparently Bartle has multiple analysis texts. I'm looking at Introduction to Real Analysis. The problem has three parts, but the first two parts not only seem easier, but are related, although the third part (the one you're asking about) doesn't seem to follow from the first two parts.

You're right, I think only regular MVT suffices to prove this. There's a lot of epsilon pushing, but I'll let you fill in most of the details.

Take b > 0 first. Start by writing out the definition of the hypothesis that f'(x) tends to b as x approaches infinity. Namely, if x is sufficiently large, say larger than M, what condition is satisfied? Apply the mean value theorem to f over the interval [M,x], and rewrite f(x)/x (algebraic manipulation) to make use of this application of the MVT. Once you have rewritten f(x)/x, you need to bound the resulting expressions.

Remember that you could always make x as large as you need or require x to be larger than multiple quantities. Also, you can always take epsilon as small as needed, since we only care about small epsilon anyways.