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in a random graph setting with $n$ vertices and edge-probability $p$ I want to show that $\binom n4p^6\to 0$ as $pn^{2/3}\to 0$. To check this is even true, I submitted this Wolfram|Alpha query.

Now why does Wolfram|Alpha calculate the limit for $n\to pn^{2/3}$ rather than $pn^{2/3}\to 0$? Is it just a misinterpretation, or is there a mathematical connection I fail to see? Surprisingly it yields the desired result ...

I also don't quite understand what Wolfram|Alpha does in the next line, and I am not even sure how to calculate the limit manually, so I'd be grateful for any insight!

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    $\begingroup$ Be careful wolframalpha is not perfect, sometimes can fail. The best thing you can do is evaluate yourself. $\endgroup$ – Masacroso Dec 13 '15 at 15:57
  • $\begingroup$ I know. It gives the right result though so I wondered if one can somehow transform $pn^{2/3}\to 0$ to $n\to pn^{2/3}$. Is there a straightforward way of evaluating such limits manually? $\endgroup$ – akkarin Dec 13 '15 at 15:59
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Use $pn^{2/3}=\varepsilon$ and then $\varepsilon\to0$.

Wolfram|Alpha

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  • $\begingroup$ Where does the $n^{-4}$ in your query come from? $\endgroup$ – akkarin Dec 14 '15 at 13:56
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    $\begingroup$ $p=\varepsilon.n^{-2/3}$, and so $p^6=\varepsilon ^6.n^{-4}$ $\endgroup$ – JonMark Perry Dec 14 '15 at 14:21
  • $\begingroup$ Ah, I see. Would you say it is obvious that the limit is $0$ as $\epsilon\to 0$, or does it need more reasoning? $\endgroup$ – akkarin Dec 14 '15 at 15:03
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    $\begingroup$ $\binom{n}{4}$ is $o(n^4)$ and $n\to\infty$ so $p\to 0$ does it $\endgroup$ – JonMark Perry Dec 14 '15 at 15:05
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    $\begingroup$ what happens otherwise? $\endgroup$ – JonMark Perry Dec 14 '15 at 15:19
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Simple answer, in wolfram alpha you can't have the product of two variables approach a value. In fact you only can have one variable approach a value. You would have to change it so one variable is approaching a certain value. If you want to keep it like you have it, then this is a job for mathematica, not wolfram alpha. See:https://mathematica.stackexchange.com/questions/21544/finding-limits-in-several-variables

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