Question: I tried evaluating the following limit with WA, but I got a wrong answer. The limit is: $$\lim\_{n\to\infty}\left|
\frac{n\sin^2(n+1)}{(n+1)\sin^2(n)}\right|$$
The result that I obtained from WA was $0$, however it's wrong. Any thoughts? How do I report this? 

You can check this by:

```
WolframAlpha[
    "Limit Abs[n Csc(n)^2 Sin(n+1)^2/(n+1)] as n goes to infinity"
]

```

> 
> 0
> 
> 
> 

$$$$
\*\*Edit: Mathematica is not able evaluate the limit, instead other computing programs like Maple and Matlab seem to do so. Why so?

Why is the equivalent expression

```
Limit[ Abs[ n Csc[n]^2 Sin[n + 1]^2/(n + 1) ], n -> ∞]

```

returned unevaluated?

Answer: 
I may not be Mathematician enough, but one can clearly see that the function given is a periodic function:

```
f = Function[ n, Abs[(n Sin[n + 1]^2)/((n + 1) Sin[n]^2)]];

Plot[ f[n], {n, 0, 10}, ImageSize -> Large, PlotTheme -> "Scientific" ]

```

[![Functionplot](https://i.stack.imgur.com/ruVJU.png)](https://i.stack.imgur.com/ruVJU.png)

For every multiple of $\pi$ the function will have a pole:

```
Limit[ f[n], n -> π] // Simplify

```

> 
> $\infty$
> 
> 
> 

And quite obviously for every value $ k\cdot\pi - 1, k \in \mathbb{Z} $ the function hits zero.

A more formal approach is discussed on Math.SE [here](https://math.stackexchange.com/a/645562/200681): Accordingly choosing $x\_n = n\cdot \pi $ and $y\_n = n\cdot \pi - 1$ will both go to $\infty$ as $n \rightarrow \infty$, but clearly then $f(x\_n) \neq f(y\_n)$ so there cannot be a limit.

**EDIT**

Take a look at $f(x) = |\csc(x)|$

```
Plot[ Abs[ Csc[x] ], {x,0,4 \[Pi] },  ]

```

[![Csc[x]](https://i.stack.imgur.com/p2Sqx.png)](https://i.stack.imgur.com/p2Sqx.png)

and that function will give an interval for the `Limit`:

```
Limit[ Abs[ Csc[x] ], x -> Infinity ]

```

> 
> Interval[{1,$\infty$}]
> 
> 
> 

But $f(x) = \csc^2(x)$ will not and simply remains unevaluated:

```
Plot[ Csc[x]^2, {x, 0, 4 \[Pi]} ]

```

[![Csc^2(x)](https://i.stack.imgur.com/IRcle.png)](https://i.stack.imgur.com/IRcle.png)

```
Limit[ Csc[x]^2, x -> Infinity ]

```

> 
> (\* remains unevaluated \*)
> 
> 
> 

While you would expect to receive the exact same result as before, wouldn't you?
Does this function have a limit? Let

```
f[n_] := Abs[n Sin[n + 1]^2/((n + 1) Sin[n]^2)] // N

```

Here we plot a few values:

```
ListPlot[f[#] & /@ Range[100000, 100200]]

```

[![enter image description here](https://i.stack.imgur.com/36oKm.png)](https://i.stack.imgur.com/36oKm.png)

It sure doesn't look like it has a limit...