Please capitalize where necessary: the existence of additive inverses for the equivalence classes is simple; i.e., the inverse of [r] is [-r].
A: The existence of additive inverses for the equivalence classes is simple; i.e., the inverse of [r] is [-r].

Q: what is this lower limit? well, the term var/n goes to zero as n increase without bound, so the average variance drops out of the picture.

A: What is this lower limit? Well, the term Var/N goes to zero as N increase without bound, so the average variance drops out of the picture.

Q: it is thus not for sceptical researchers who seek purely scientific analyses.
A: It is thus not for sceptical researchers who seek purely scientific analyses.

Problem: netherlands antilles guilder 176.380 naf
Can you repeat this sentence, but capitalize it?
Answer: Netherlands Antilles Guilder 176.380 NAF

Text: the white chocolate is mixed with licorice and chili (it is the crazy good) but in a slightly different relationship than the one i made earlier der er vist mere lakrids og en smule mindre chili i denne.

Correct capitalization: The white chocolate is mixed with licorice and chili (it is the crazy good) but in a slightly different relationship than the one I made earlier der er vist mere lakrids og en smule mindre chili i denne.

Text: these rooms cannot accommodate an extra bed, but can accommodate one cot for a baby.

Correct capitalization:
These rooms cannot accommodate an extra bed, but can accommodate one cot for a baby.