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The Volume conjecture says that if $J_n(q)$ are the colored Jones polynomials of a knot $K \subset S^3$, then $$\lim_{N \to \infty} \frac{ 2 \pi}N \left\vert J_N(e^{2\pi i / N})\right\vert = vol(K)$$ Here we are assuming that the complement of $K$ is hyperbolic, and $vol(K)$ is its hyperbolic volume. (The normalization is that $J_N(q) = 1$ for the unknot.) I have an ignorant question about this:

Q: Is it obvious that this limit exists?

One thing that might be helpful for showing the limit exists is that the $J_n(q)$ satisfy a (generalized) recursion relation (this is a theorem of Garoufalidis and Le). More precisely, there are Laurent polynomials $a_i(-,-)$ such that $$ \sum_i a_i(q,q^N) J_{N+i}(q) = 0$$ (This is true for any $N$, and the $a_i$ don't depend on $N$.) But the existence of this recursion relation was proved several years after the volume conjecture.

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No, this is unknown. There are heuristic arguments for convergence based on the stationary phase approximation, but as far as I know, no one has made the argument precise in general. The closest I know of to a proof of convergence is an upper bound on the limsup in terms of the crossing number given in Theorem 1.3 of this paper by Garoufalidis and Le, although I should add the caveat that I haven't been following the developments on this topic closely recently.

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