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This is the example I tried to solve with WolframAlpha. The result is also in the picture. Can someone explain me what the limes of this example is and how can i get there by hand?

Limit[Power[(40)Divide[(40)1-Power[n,2](41),Power[n,2]-4](41),n],n->∞]

OttoFran
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  • I would guess : a limit cycle between $e^{2 i 0}$ and $e^{2 i \pi}$ (the ratio is nearly $-1$) – Raymond Manzoni Nov 17 '21 at 20:55
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    @RaymondManzoni: Running the equivalent code in Mathematica seems to verify your guess; it returns E^(2 I Interval[{0, \[Pi]}]), which I think should be interpreted as "all points approach the unit circle in the complex plane." But that doesn't make a lot of sense given that the sequence is real. – Michael Seifert Nov 17 '21 at 21:02
  • see this reference about adherence points: https://math.stackexchange.com/q/3512083 – Jean Marie Nov 17 '21 at 21:15

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I would have understood if Mathematica had given this answer

$$\text{the would-be limit is} \ e^{i0}=1 \ \text{and/or} \ e^{i\pi}=-1$$

which are the 2 so-called adherence points to the sequence, not the limit in fact because strictly speaking the limit doesn't exist.

Indeed the content of the parenthesis when $n \to \infty$ is equivalent to $\frac{-n^2}{n^2}=-1$, therefore our sequence is equivalent to $(-1)^n$, somewhat "constantly hesitating" between $-1$ and $+1$...

The conclusion : you are in GIGO context (Garbage In, Garbage Out): when faced to a nonsense situation the best software (?) on Earth can give you nonsense answers...

Jean Marie
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  • Hi Jean Marie glad to meet you again! Mma likes complex numbers and probably replaced the approximate $-1$ by $e^{i \pi}$. For large reals $n$ he then got $\displaystyle,e^{n i \pi},$ so that, not restricting $n$ to integer values, produced the whole complex unit circle indicated by Michael Seifert : $\exp(2,i \operatorname{Interval}[{0, \pi}])$ (unable to provide a single value he will produce an interval). Cheers, – Raymond Manzoni Nov 17 '21 at 21:52
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    Mathematica's "brain" has been pushed to its limits here... Your explanation is very likely to be the good one... Have a good time till the end of the year... – Jean Marie Nov 17 '21 at 21:59
  • and beyond! :-) All the best, – Raymond Manzoni Nov 17 '21 at 22:00
  • @Raymond Manzoni And beyond, of course ! Surfing on the n-th COVID wave... – Jean Marie Nov 17 '21 at 22:04