set theory - Which very large cardinals are preserved under Woodin's forcing for $mathsf{AC}$?

Solving restrict with out L’hopital $lim_{xto0} left(left(frac{xe^x}{e^x-1}-1right)/xright)$ Answer

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Solving restrict with out L’hopital $lim_{xto0} left(left(frac{xe^x}{e^x-1}-1right)/xright)$

I’m making an attempt to decipher this restrict utilizing transformations & widespread limits however I’ve been unable to:
$$lim_{xto0}left( frac{frac{xe^x}{e^x-1}-1}{x}privilege)$$

This restrict simplifies to $1/2$ utilizing L’Hôpital’s rule however I needed to employ it 3 instances in organize to get the outcome. (I’m making an attempt to show that the duty is differentiable at $x=0$.)

So if anybody is ready to decipher it utilizing transformations solely with out L’hopital I might breathe grateful.

Thanks to your time and remain favorable…

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