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I would like to know if there is any non trivial function $f(x)$ and a $x_0$ such that $$\lim_{x\to\ x_0} f(x)$$
is currently not known, with $x_0 \in \mathbb{R}\cup \{-\infty, +\infty \}$.
An example of a "trivial" function is $A(x)$ where $A(x)$ denotes the number of perfect numbers not greater than $x$. It is an open problem to find the value of $\lim_{x\to\infty} A(x)$, since we don't know if there are infinitely many perfect numbers.
I would prefer a limit which can be recognized by a high school student.

Ed Pegg
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Konstantinos Gaitanas
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    A slightly different example would be a function counting the integers $<x$ for which the sequence in the Collatz conjecture doesn't end at $1$ (I call it slight different because wee don't know if any such integer exist, i.e. whether the function ever becomes $\neq 0$, where we know 50 perfect numbers). But that is probably also trivial, but you haven't given us a definition of trivial that is actually workable. – Henrik supports the community Jul 24 '18 at 08:08
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    The value of $$\lim_{n\to\infty}R(n,n)^{\frac1n}$$ where $R(n,n)$ is a so-called Ramsey number is unknown. It is known that the limit (if it exists) lies in the interval $[\sqrt2,4].$ – bof Jul 24 '18 at 08:42
  • It is unknown whether $1/(n^2\sin n)$ converges as $n \to \infty$ (see Are there any series whose convergence is unknown?). Not sure if it is duplicate since it asks for series, but one of the answer gives this sequence as an example, so in a sense... – Sil Aug 18 '18 at 00:13

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Brun's theorem states that the sum of reciprocals of twin primes is convergent, but there is no other known expression for the limit.

Ludvig Lindström
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