Borzki,

In fact to skip the math trick in your original question you must start with F (action and cause of the deformation).
By math a finite F=k*x for case k-->infinity (because l-->0) requires x-->0 to generate in the right member of equation an indeterminate case of limits (as to be solved giving a finite result).
I know it's sounds like a joke, but the interpretation- linked to continuous medium hypothesis- is : in linear domain, a finite force will generate almost no deformation to a bar than tends to disappear! And, as I've written previously, when the bar physically disappeared, no bar and no deformation are there, that's why there isn't any singularity.