How to solve the following limit $\lim_{n\to\infty} n^2\cdot(\sqrt[n]{x}-\sqrt[n+1]{x})$?

Question

$$\lim_{n\to\infty} n^2\cdot(\sqrt[n]{x}-\sqrt[n+1]{x})$$

My first idea, since it's $0 \cdot \infty$ (might be wrong though) was to re-write the multiplication as a fraction $\frac{\sqrt[n]{x}-\sqrt[n+1]{x}}{\frac{1}{n^2}}$ and apply L'Hospital but I don't think it gives the right answer (wolfram says it's ln x) so I'm pretty much out of ideas since I don't see any remarkable limit there

Thanks in advance.

By the way, I found the problem in a book that's for eleventh grade (or junior year) so I'm pretty sure there's a trick somewhere that I'm not aware of

@MathematicianByMistake Formally it is not correct, but if you think in $n$ as a real variable, the function is continuous. — ajotatxe, Apr 28 '19 at 10:29
@ajotatxe I am wondering where the mistake is in the application of L'Hospital. Hopefully OP will show what he tried. — MathematicianByMistake, Apr 28 '19 at 10:31
Related Find the limit $\lim_\limits{n\to{\infty}}n^2(\sqrt[n]{x}-\sqrt[{n + 1}]{x})$ and my inquiry about Stolz-Cesàro theorem. Also see Find the Limit (n to infinity) — Sil, Apr 28 '19 at 10:36
@Sil, it's not just related, it looks like a proper dupe. Any reasons for not raising as a duplicate? — rtybase, Apr 28 '19 at 10:44
@rtybase The first one actually asks about Cesaro-Stoltz application, the second one looks like a dupe but the quality seems to be bad. But feel free to mark it that way... — Sil, Apr 28 '19 at 10:46
@Sil, no, no ... the privilege is yours. I will support the vote though. — rtybase, Apr 28 '19 at 10:50
@rtybase I've retracted it already, so I cannot anymore on this one. It's okay, it''s not like we get any reputation for marking duplicates... — Sil, Apr 28 '19 at 10:52
@Sil you never know, all these things add up, I can tell. Besides, it's our (as part of the community) duty to keep it clean and help others (in this case: promote old questions, help new joiners to look for answers before asking). — rtybase, Apr 28 '19 at 10:55

How to solve the following limit $\lim_{n\to\infty} n^2\cdot(\sqrt[n]{x}-\sqrt[n+1]{x})$?

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