Do black hole singularities actually merge? new questioner here so please be gentle.

Do the singularities of merging black holes actually merge together?  I mean there are 2 infinitely small objects (singularities) that are trying to find each other but for them to merge they have to join at the EXACT same point in space.  I know they can get close, but to expect them to align perfectly with zero tolerance seems impossible.
Infinities are hard to bend the mind around, but in this case, the merge is not impossible. Yes, the distance between them must reach zero in order to make the black holes merge, but the rate of which energy is lost to gravitational waves also increases when they get close to each other.

We are therefore dealing with a $\frac{\mathrm{infinity}}{\mathrm{infinity}}$ problem, where a finite limit may, and in this case does exist.

Do not forget your $\lim$'s!