I'm seeing this behavior in sage 6.4, just built from git:
$ ./sage
┌────────────────────────────────────────────────────────────────────┐
│ Sage Version 6.4, Release Date: 2014-11-14 │
│ Type "notebook()" for the browser-based notebook interface. │
│ Type "help()" for help. │
└────────────────────────────────────────────────────────────────────┘
sage: e = SR('a*f(x)+g(x)')
sage: e
a*f(x) + g(x)
sage: e.limit(x=0)
limit(a*f(x) + g(x), x, 0)
sage: e.limit(x=0).limit(a=0)
limit(g(x), x, 0)
sage: de = diff(e,SR('x'))
sage: de
a*D[0](f)(x) + D[0](g)(x)
sage: de.limit(x=0)
limit(a*D[0](f)(x) + D[0](g)(x), x, 0)
sage: de.limit(x=0).limit(a=0)
D[0](g)(x)
The last result should be limit(D[0](g)(x), x, 0).
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