Limit not evaluating

Question

Why is the input Limit[IntegerPart[Sin[x]/x], x -> 0] not being evaluated?

Answer 1

A closely related problem was treated in this question I asked some time ago that received a beautiful answer from @acl.

The documentation for IntegerPart[] says:

Mathematical function, suitable for both symbolic and numerical manipulation.

Nevertheless, the symbolic part of the assertion has been proved false in the aforementioned question. Just take a look at what Mma thinks of its derivative:

So, What does Limit[] do when a non-symbolically treatable function is given as an argument? Let's try it:

f[x_?NumericQ] := Sin[x];
Limit[f[x], x -> 8]
(*
  Limit[f[x], x -> 8]
*)

Ha! it does nothing!

That doesn't mean that Mma can't calculate limits for analytic non-symbolically treatable functions. It can:

f[x_?NumericQ] := Sin[x];
Limit[f[x], x -> 8, Analytic -> True]
(*
Sin[8]
*)

Of course this is useless in your case since IntegerPart[] is not analytic.

So, there is a deadlock: Mma does not know how to treat IntegerPart[] symbolically, and also doesn't know to calculate limits of functions that are only numerically valued and not analytical.

Sorry :)

Edit

This is a cheater using PiecewiseExpand:

f[x_] := Assuming[ FindMaxValue[Sin[x]/x, x] <= Sin[x]/x <= FindMinValue[Sin[x]/x, x],
                   PiecewiseExpand@IntegerPart[Sin[x]/x]];

Off[FindMaximum::lstol]
Limit[f[x], x -> 0]
On[FindMaximum::lstol]

End Edit