TITLE: Find the limit $\lim_\limits{n\to{\infty}}n^2(\sqrt[n]{x}-\sqrt[{n + 1}]{x})$ and my inquiry about Stolz-Cesàro theorem
QUESTION [6 upvotes]: Let $x>0$, find the limit
$$\lim_\limits{n\to{\infty}}{n^2\left(\sqrt[n]{x}-\sqrt[{n + 1}]{x}\right)}$$
I use Maclaurin series and find out that the limit is $\ln x$.
And this is the answer I get from a math forum:
"Let $x_n=\sqrt[n]{x}-1$, $y_n=\dfrac{1}{n}$, then $(x_n)\to0,(y_n)\to0,(y_n)\downarrow$ when $n\to\infty$.
Use Stolz-Cesaro theorem, we have:
$$\mathop {\lim }\limits_{n \to \infty } \dfrac{{\sqrt[n]{x} - \sqrt[{n + 1}]{x}}}{{\frac{1}{n} - \frac{1}{{n + 1}}}} = \mathop {\lim }\limits_{n \to \infty } \dfrac{{\sqrt[n]{x} - 1}}{{\frac{1}{n}}} = \ln x$$
Then $\mathop {\lim }\limits_{n \to \infty } {n^2}\left( {\sqrt[n]{x} - \sqrt[{n + 1}]{x}} \right) = \mathop {\lim }\limits_{n \to \infty } \dfrac{{{n^2}}}{{n\left( {n + 1} \right)}}\cdot\dfrac{{\sqrt[n]{x} - \sqrt[{n + 1}]{x}}}{{\frac{1}{n} - \frac{1}{{n + 1}}}} = \ln x$.
/END"
I just wonder what form Stolz-Cesaro theorem that he used, because I just learn Stolz-Cesaro theorem when $(y_n)\to\infty$.
Could anyone help me to get this? Thank you in advance.

This is "another form of Stolz theorem" in the forum that I mentioned: Let $(x_n),(y_n)$
    be two sequences of real numbers. Assume $\mathop {\lim }\limits_{n \to \infty } {x_n} = \mathop {\lim }\limits_{n \to \infty } {y_n} = 0$; $y_n>0$, $(y_n)$ is strictly decreasing  and the following limit exist $\mathop {\lim }\limits_{n \to \infty } \dfrac{{{x_{n + 1}} - {x_n}}}{{{y_{n + 1}} - {y_n}}} = L$. Then $\mathop {\lim }\limits_{n \to \infty } \dfrac{{{x_n}}}{{{y_n}}} = L$. Is this correct?

REPLY [2 votes]: The version of Cesaro-Stolz you mention is correct, but it's application for the problem at hand is wrong. The theorem is used to deduce the limit of $a_n/b_n$ from that of $(a_n-a_{n+1})/(b_n-b_{n+1})$ and not the other way round. The right approach to solve the problem is given by user "lab bhattacharjee" in his answer.