English or French translation of a paper in German

Question

Is there an English or French translation of the following paper (or at least a work in these languages that summarize its main results)?

Stolz, O. "Ueber die Grenzwerthe der Quotienten." Math. Ann. 15, 556-559, 1879.

Answer 1

The paper is quite short, so I’ve simply summarized it here. It’s an addendum to a paper by Stolz with the same title in vol. $14$, pp. $231$-$240$. He says that when he wrote the earlier paper he missed a relevant note by V. Rouquet. Rouquet, he says, starts from the following lemma (in which I’ve modernized the notation and some of the terminology):

Lemma. If $F(x)$ is continuous at all finite $x>x_1$ and equipped with a derivative $F'(x)$ that has a limit as $x\to+\infty$, then the fraction $F(x)/x$ also has a limit as $x\to+\infty$, given by $$\lim_{x\to+\infty}\frac{F(x)}x=\lim_{x\to+\infty}F'(x)\;.$$

In a footnote he remarks that although Rouquet’s proof assumes that $\lim F'(x)$ is finite, the lemma holds when $\lim_{x\to+\infty}F'(x)=+\infty$ as well and gives a proof of this.

He notes that the lemma does not require that $F(x)$ itself have a limit as $x\to+\infty$, offering the example $F(x)=\sin\sqrt{x}$. He says that Rouquet then goes on to infer that if $y=f(x)$ and $z=\varphi(x)$ are continuous functions that both tend to $+\infty$ as $x$ approaches a finite $a$ or tends to $+\infty$, one can regard $y$ as a function of $z$, and since

$$\frac{dy}{dx}=\frac{f'(x)}{\varphi'(x)}\;,\tag{1}$$

we have the following result: if under these hypotheses $\lim\frac{f'(x)}{\varphi'(x)}$ exists, then so does $\lim\frac{f(x)}{\varphi(x)}$, and the two limits are equal.

Stolz then points out that this is not always true. For example, if $f(x)=x+\sin x\cos x$ and $\frac{\varphi(x)}{f(x)}=e^{\sin x}$, then $\lim_{x\to+\infty}f(x)=\lim_{x\to+\infty}\varphi(x)=+\infty$, and

$$\frac{f'(x)}{\varphi'(x)}=e^{-\sin x}\cdot\frac{2\cos x}{x+\sin x\cos x+2\cos x}\longrightarrow 0\quad\text{as}\quad x\to+\infty\;,$$

but $\frac{f(x)}{\varphi(x)}=e^{-\sin x}$ oscillates over the range between $\frac1e$ and $e$ and has no limit as $x\to+\infty$.

Therefore, Stolz says, one can’t use formula $(1)$ unconditionally. A natural requirement for using it is the assumption that there is a $z_1$ such that $y$ is a continuous function of $z$ for all finite $z\ge z_1$ or for all finite $z\le z_1$. This, so far as he knows, can be assumed only when $x$ is a continuous function of $z$ for these values of $z$. But $\varphi$ has a continuous inverse $\psi$ only on intervals $(a-\delta,a)$ or $(a,a+\delta)$ (where $\delta>0$), and $(x_1,+\infty)$ or $(-\infty,x_1)$, on which $\varphi$ is monotonic. Moreover, $(1)$ is meaningful only when $f'(x)$ and $\varphi'(x)$ are not simultaneously $0$ or infinite. He then states a theorem that actually does follow from $(1)$ and Rouquet’s lemma:

Theorem. Let $f(x)$ and $\varphi(x)$ be functions that are real-valued, continuous, and differentiable on one of the four intervals mentioned in the previous paragraph and are such that $f'(x)$ and $\varphi'(x)$ are not simultaneously $0$ or infinite. Suppose further that at least one of $f$ and $\varphi$ is monotonic on the interval and tends to $+\infty$ or $-\infty$ as $x\to a^-$, $x\to a^+$, $x\to+\infty$, or $x\to-\infty$ according to the type of interval. Then if $\lim\frac{f'(x)}{\varphi'(x)}$ exists, so does $\lim\frac{f(x)}{\varphi(x)}$, and the two limits are equal.

He adds that if only $f(x)$ satisfies the conditions, e.g., if $\lim\varphi(x)$ is finite, then the signs of the infinite limits $\lim\frac{f'(x)}{\varphi'(x)}$ and $\lim\frac{f(x)}{\varphi(x)}$ can differ.

Moreover, it is not necessary that both functions $f(x)$ and $\varphi(x)$ have limits; an example is given by $f(x)=\sin x$, $\varphi(x)=x^2$, $\lim_{x\to+\infty}\frac{f(x)}{\varphi(x)}=\lim_{x\to+\infty}\frac{f'(x)}{\varphi'(x)}=0$.

This theorem, he says, agrees with that of du Bois-Reymond on p. $502$ of vol. $14$. He notes that it is also not difficult to derive it using his methods, so that it replaces the fourth theorem (on p. $238$) of his earlier paper. This is because the second theorem of that paper (on p. $234$) also holds when of the continuous functions $f(x)$ and $\varphi(x)$ only the latter is monotonic on $(x_1,+\infty)$ and therefore has an infinite limit as $x\to+\infty$; in a footnote he gives the necessary modifications to the proof.

Finally, he points out that although the theorem is easily derived from Rouquet’s lemma, without any need for the lemmas in §$1$ of his earlier paper, those lemmas are nevertheless of some importance and cannot be derived from the theorem without the assumption that $\lim_{x\to+\infty}\frac{f'(x)}{\varphi'(x)}$ exists.

In response to an observation by du Bois-Reymond, he concludes with a footnote justifying the observation on p. $232$ of his earlier paper that a continuous function $f(x)$ that has an infinite limit as $x$ approaches some limiting value actually has either $+\infty$ or $-\infty$ as its limit.