Meaning of "Предельный"

Question

The dictionary gives the meanings marginal and maximum for предельный, which are practically opposites. Consequently I don't know how to interpret it in the following passage:

Не утруждая себя приветствием, он с предельной категоричностью заявил:

– Эй вы! Имейте в виду, я ни на секунду не считаю себя вашим пленником.

How would a native interpret this?

Answer 1

I believe utmost more fits your example:

Not bothering to greet anyone, he announced with the utmost rigidity:
- Hey you! Don't think for a second that I'm going to be your prisoner!

Answer 2

OK, I'll try to let you see how the word might make sense.

Start with a "предел"=limit.

Then предельный = taken to its limit = extreme. The quality pronounced as strong as it possibly could

• (of course, there is also a more broad adjectival meaning "limit-related", extensively used in mathematics — for instance, "предельное значение" = limiting value/the value of a limit , "предельный переход" = limit transition, transition to a limit etc.)

Now you can think of it in two ways:

• limit = go as far as you can → предельный = utmost
• matematical limit of some value when ∆x → 0: предельный = marginal (showing how a function changes when you introduce an infinitesimal change to its variable)

Honestly, I do not know why would you chose the second meaning when there is clearly no calculus or economics involved :). Probably, the dictionaries should include some examples of very specific uses like these. Sometimes it takes me ages to sift through the minor meanings I would hardly ever need (that's why I love oxfordictionaries, with their elegant hierarchy of meanings and ready-to-use examples)

Now, back to your sentence. "Категоричный" means the opposite of being soft, tolerant, indirect and ready to change your opinion. So here's how the meaning would be understood:

• He didn't even trouble himself with a greeting and said most adamantly: ...