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What is the domin of f(x)= xlnx.

I know that: domin of x is R and domin of lnx is x greater than 0 so the domin of them will be x greater than 0

but when I subtitude 0 on f(x) I find (o.[tex]\infty[/tex]) and this form from LHopital Rule

so what shall I do I use the first tip or I use LHopital Rule to find the domin?