Strong versus weak coupling expansion to solve hard problems

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For the quintic equation

$$x^5 + x = 1$$

it can be seen that when taking the strong coupling limit to solve

$$x^5 + \epsilon x = 1$$

perturbatively, summing the terms of all orders in $\epsilon$ converges to the right result (which is inside the radius of convergence), whereas taking the weak coupling limit and expanding

$$\epsilon x^5 + x = 1$$

the result of summing all terms lies far outside the radius of convergence.

How can I generally see which approach I have to take in advance (where to put the $\epsilon$) when looking at such problems? Does it always make a difference such that only the strong or weak coupling case converges to the right result, or can there be situations where one obtains the right finite result in both cases (some kind of S-duality)?

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