Are there limits of which we're not able yet to find the value or not even prove non/existence? I really like working out limits, so I've been wondering:

Are there limits we're struggling to evaluate? Are there limits of which we're not succeeding in proving the existence or nonexistence?


Going similarly to @JessicaK's answer, there are plenty of other infinite series which are open. These are certainly limits, since
$$\sum_{n=1}^\infty x_n \equiv \lim_{N\to\infty}\sum_{n=1}^N x_n$$
by defintion.
A famous infinite series which we are unable to evaluate at the moment is
$$\sum_{n=1}^\infty \frac{1}{n^3\sin^2 n}\tag1$$
There's actually a MO question about this series, and no resolution is given there.
In an answer that very question, the above series is shown to be related to the "difficult limit" o
$$\vert n\sin n\vert,n\to\infty$$
Said answer ends up non-trivially relating this limit to the convergents of $\pi$. Note that the limit itself clearly goes to infinity ($\vert n\sin n\vert = \vert n\vert\vert \sin n\vert$), however investigating the behavior of the function as it tends to infinity is sort of the "next level" past just evaluating the limits.
