Limit not evaluating

Why is the input Limit[IntegerPart[Sin[x]/x], x -> 0] not being evaluated? • No idea. Limit[IntegerPart[Sinc[x]], x -> 0] works, though. – J. M. will be back soon May 16 '12 at 17:07
• In general finding IntegerPart requires infinite precision, which may be a little much to ask :-). Nevertheless MMA should be able to figure this one out from the series, and in fact it quickly gets the correct answer with, say, Limit[IntegerPart[Series[Sin[z]/z, {z, 0, 4}]], z -> 0]. But this is a snare: replacing the 4 with 1 gets a wrong answer. It does not seem to realize that $1+O[z]^2$ can be less than $1$! – whuber May 16 '12 at 19:43
• Limit[IntegerPart[Sin[x]/x], x -> 10^-80] returns 0, so it must be a precision thing. – Guillochon May 16 '12 at 22:25
• @J.M. I get 0 for Limit[IntegerPart[Sinc[x]], x -> 0]. Which version are you running? I'm using v8.0.4, but it does the same on v7. – rcollyer May 17 '12 at 1:45
• In:= Limit[IntegerPart[Sin[x]/x], x -> 0] Out= 0 (Coming in version 9.) – Daniel Lichtblau Jun 12 '12 at 15:38

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
*)

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]