Manually rewriting the root does the trick:

In [6]: l = Limit((-(6 - x)**(Integer(1)/3) + 2)/(x + 2),x,-2,'-'); l
Out[6]:
      ⎛    3 _______⎞
      ⎜2 - ╲╱ 6 - x ⎟
 lim  ⎜─────────────⎟
x─→-2⁻⎝    x + 2    ⎠

In [7]: _.doit()
Out[7]: 1/12

So the piecewise expression returned by real_root is not quite amenable to all operations, in particular derviatives and sign don't seem to mix well. For reference:

In [8]: real_root(x - 6, 3)
Out[8]:
⎧3 _________
⎪╲╱ │x - 6│ ⋅sign(x - 6)  for im(x) = 0
⎨
⎪       3 _______
⎩       ╲╱ x - 6            otherwise

I think the underlying issue is has to do with this: if x is real computing the diff is no problem:

In [16]: diff(sign(x-8))
Out[16]: 2⋅δ(x - 8)

In [17]: diff(sign(x-8)).subs(x, 0)
Out[17]: 0

But if I set x to be real before calling limit, this assumption is lost (or discarded on purpose?) when the Dummy in the diff.subs is created. And then we're left with the behavior for a general complex variable:

In [18]: diff(sign(y-8))
Out[18]:
d
──(sign(y - 8))
dy

In [19]: _.subs(y, 0)
Out[19]:
⎛d              ⎞│
⎜──(sign(y - 8))⎟│
⎝dy             ⎠│y=0

Since I'm not at all familiar with this part of the codebase that's as far as I'm going with this.

Uh oh!

Assume real root in limit #18363

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