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->∞]
,Power[n,2]-4](41),n],n->∞]](https://i.stack.imgur.com/hc8AJ.png){kind=link}
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...