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When parameters to a integral are integers, I often fall back to calculating the "general" result, make a table where the parameter is explicitly given integer values, and then feed to FindSequenceFunction. In this case: table = Table[h[x, a, e] - g[x, a, e], {a, 1, 11}]; limits = Limit[#, e -> 0] & /@ table; f = FindSequenceFunction[limits]; ...

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