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I have noticed that the words ultima and proxima are rarely used.

How do native speakers use these?

"Ultimate", "penultimate", "proximate", and "approximate" are English words.

"Ultimate" is common, partly because it is used in advertising.

"Penultimate" is less common than "ultimate", partly because nobody wants to advertise that their product is second-best.

"Approximate" is common, both as an adjective and as a verb. The adjective "approximate" is more formal than the adjective "rough"; the verb "approximate" is more formal than "make a careful guess" or "come close to".

"Proximate" is not very common. Both "penultimate" and "proximate" are sometimes used by educated writers.

Ultima and proxima are so rare that they might as well still be Latin words. Until I looked them up just now, I had no idea that ultima specifically referred to the last syllable in a word, instead of generally the "furthest out" thing in a group of things. The most common use of ultima is in the title of the on-line game Ultima Online. The most common use of proxima is in the name of the second-closest star to the Earth, Proxima Centauri.

Those both exist as words in English, but with very specialised meanings or very infrequent use. These relate to their meanings in Latin, which are reflected in modern Spanish (and possibly Portuguese, too), albeit with accents on some letters.

Ultima is used in specialised situations to refer to the last syllable of a word.

Proxima is a little-known term that means "of (the) next month", for example "on the 3rd proxima" means "on the third day of next month".

These are both rather obscure words and most people are not familiar with them. I only know what they mean now because I thought I should look them up before saying "those are not English words".

Don't use them. Well, use ultima if you're in the sort of context where other people use it. Don't use it otherwise, and basically no-one uses proxima unless they're trying to show off their expansive vocabulary.

I have noticed that the words ultima and proxima is rarely used.

How native speakers use these?

I know from my knowledge of classical Latin and of Spanish that these words mean 'last' and 'next' respectively.

How do native speakers use these? They don't. I have never used or heard or read these words in English.

Edit 1

I see from @SamBC that these words appear in English dictionaries with very specialised meanings. I didn't know those meanings and I will not make any attempt to remember them.

Edit 2

It occurs to me that astronomers use 'proxima' as in Proxima Centauri, a red dwarf star, in the constellation of Centaurus. This is however the name of the star and is not used in general conversation by most people unless they happening to be discussing that particular celestial body.

In 18th and 19th century business correspondence, these words, invariably abbreviated as "Ult." and "Prox." were often used to mean mean 'of the last month" and "of the next month". Such a letter might run:

In your favor of the 19th ult., you proposed delivery on the 20th prox. The supplies are needed by the 10th prox. at the latest.

(Here "favor" was a polite term for "letter addressed to me" now also obsolete.)

I have not seen this usage in any document later than about 1920, and rarely in any later than 1880. Currently I see them only in historical fiction of that period, or in the star name Proxima Centauri mentioned in other answers.