# Transform an expression to remove the singularity

I have the expression Sin[c x]/c. Clearly it is undefined at $c=0$, but that is merely a removable singularity, since $$\frac{\sin(cx)}c=x\frac{\sin(cx)}{cx}= x\operatorname{sinc}(cx),$$ which is continuous everywhere. Is it possible to get this sort of transformation automatically in Mathematica using the built-in formula manipulation tools? I tried a few things like FullSimplify, TrigReduce, etc. but nothing helped.

I'd like to be able to do this for (1 - Cos[c x])/c, too; that is, automatically transform it into a form that has no singularity at $c=0$.

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You can go backwards: FunctionExpand[x Sinc[c x]] –  bill s Feb 20 at 19:04
@bill: I suppose what I need is a FunctionReduce along the lines of TrigExpand/TrigReduce. :) –  Rahul Feb 21 at 7:06
I can't even find a way to go back from Sin[c x]/(c x) to Sinc[c x]. One would think there should be some kind of ComplexityFunction that could be used with FullSimplify to achieve this. –  bill s Feb 21 at 16:39