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Sometimes when trying to find the limit of a function $f(n)$ when $n\to\infty$ using Limit[f,n->Infinity], it seems that Mathematica is unable to compute it numerically. Instead, I get a result like

Limit[f,n->Infinity]

in place of a numerical value. I'm guessing it is unable to simplify the expression enough to be able to compute a value. In this case it would be helpful at least to know if the limit exists, but I haven't found a way to determine this after reading the Mathematica documentation.

My questions are:

If Mathematica returns the original expression when using Limit[], does this mean it is unable to compute the limit, or am I missing something?

How may I determine the existence of a limit in Mathematica, when it is not possible to determine a value for it?

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Mathematica is not always able to calculate limits and sometimes not only gives up, but makes mistakes.

For example, writing:

Limit[n Sin[2 Pi E n!], n -> Infinity]

we get:

Indeterminate

when, in reality, the limit exists and is valid 2 Pi.

On the other hand, by writing:

DiscreteLimit[n Sin[2 Pi E n!], n -> Infinity]

we get the correct result:

$2 \pi$

A last chance that could sometimes be comfortable is the following:

Limit[(1 - Cos[x])/x^2, x -> 0]
Needs["NumericalCalculus`"]
NLimit[(1 - Cos[x])/x^2, x -> 0, WorkingPrecision -> 15]

through which we obtain:

$\frac{1}{2}$

0.500000

I hope it is useful.

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    $\begingroup$ Limit[n Sin[2 Pi E n!], n -> Infinity] doesn't exist, so Indeterminate is no mistake $\endgroup$ – Coolwater Jul 7 '18 at 9:34
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    $\begingroup$ To reinforce comment by @Coolwater, look at LogLinearPlot[n Sin[2 Pi E n!], {n, 1, 100}, WorkingPrecision -> 200, PlotRange -> All] $\endgroup$ – Bob Hanlon Jul 7 '18 at 13:09
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    $\begingroup$ The limit exist if $n\in\mathbb N$, and it doesn't if $n\in\mathbb R$ (cf. the code by Bob Hanlon vs. Show[DiscretePlot[n Sin[2 Pi E n!], {n, 1, 30}, WorkingPrecision -> 100], Plot[2 Pi, {n, 1, 30}]]). The output my MMA is correct: Indeterminate in the real case, 2 Pi in the discrete case. $\endgroup$ – AccidentalFourierTransform Jul 7 '18 at 13:31

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