I'm working on another analysis problem: Let $f$ be a differentiable function on an interval of the form $(a,+\infty)$. Prove that if there is a number $r > 0$ such that $\lim_{x\to\infty}(rf′(x)+f(x))=L$ is finite, then $\lim_{x\to\infty}f′(x)=0$ and $\lim_{x\to\infty}f(x)=L$.

I figured out the proof in the case where $r>0$ and $\lim_{x\to\infty}f(x)$ exists and is nonzero and the case where $r<0$ and $\lim_{x\to\infty}f(x)$ exists and is finite (it's a pretty simple application of L'Hôpital's Rule, once you establish it applies). However, I can't seem to be able to show that $\lim_{x\to\infty}f(x)$ must exist. I also don't know where to go with the zero case for $r>0$ and the infinite case for $r<0$.

If it helps, this is in the section on L'Hôpital's Rule in the chapter on derivatives in Joseph L Taylor's Foundations of Analysis. Any help will be well appreciated.


There is a simpler answer that comes from a well-documented trick. See that \begin{align*} \lim_{x\to \infty} f(x) &= \lim_{x\to \infty} \frac{e^{x/r}f(x)}{e^{x/r}} \\ &= \lim_{x\to \infty} \frac{e^{x/r}(f'(x)+f(x)/r)}{e^{x/r}/r}\text{ (l'Hôpital)} \\ &= \lim_{x\to \infty} \frac{e^{x/r}(rf'(x)+f(x))}{e^{x/r}} \\ &= \lim_{x\to \infty} [rf'(x)+f(x)] \end{align*} This trick also works in the solution that uses differential equations, but it is done implicitly here. There is a chart here that justifies the use of l'Hôpital.

To solve this problem using differential equations theory, write $$rf'(x)+f(x) = L+\epsilon(x)$$ where $\lvert \epsilon(x)\rvert$ is vanishingly small as $x\to \infty$. The solution to this ODE with initial condition $f(x_0) = c$ is $$f(x) = e^{-x/r}\left(ce^{x_0/r}+\int_{x_0}^x \frac{L+\epsilon(\xi)}{r}e^{\xi/r}\,\mathrm{d}\xi\right)$$ Then, l'Hôpital and the fundamental theorem of calculus give us $$\lim_{x\to \infty} f(x) = \lim_{x\to \infty} \frac{ce^{x_0/r}+\int_{x_0}^x \frac{L+\epsilon(\xi)}{r}e^{\xi/r}\,\mathrm{d}\xi}{e^{x/r}} = \lim_{x\to \infty} \frac{\frac{L+\epsilon(x)}{r}e^{x/r}}{e^{x/r}/r} = \lim_{x\to \infty} [L+\epsilon(x)] = L$$ From this, we can conclude that $\lim_{x\to \infty} f'(x) = 0$.

  • $\begingroup$ The problem I have with this approach is that, since this is an introductory analysis course, we shouldn't be using even calculus unless it's a result that's already been proven earlier in the course, let alone differential equations. $\endgroup$ – themathandlanguagetutor Sep 1 '17 at 6:01
  • $\begingroup$ In addition, even just showing that the function satisfies the differential equation, I don't think I'd have the tools to show that L'Hôpital's Rule applies. $\endgroup$ – themathandlanguagetutor Sep 1 '17 at 6:07
  • $\begingroup$ It's not hard to show that either both the numerator and denominator have infinite limits or both go to $0$ (depending on the sign of $r$), although it's a bit of a moot point if this solution is too far removed from the intended method. How much calculus do you have at this point? Can you use the fundamental theorem? $\endgroup$ – Michael Lee Sep 1 '17 at 6:11
  • $\begingroup$ pretty slick proof though--not gonna lie. $\endgroup$ – Rustyn Sep 1 '17 at 6:16
  • $\begingroup$ We are allowed to use derivatives of elementary functions and the fact that they're continuous on their natural domains. We've defined limits (including one-sided and infinite) derivatives, and we've established the basic rules of how to use them (such as linearity, product rule, etc.) We've also established the Mean Value Theorem and Cauchy's Mean Value Theorem, as will as several involving continuity and uniform continuity. We also have quite a few theorems regarding sequences. What we do not yet have is integrals; we haven't even defined them (in this course). $\endgroup$ – themathandlanguagetutor Sep 1 '17 at 6:18

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