Learn English – “it is too broad in asking about ‘a limitless’/’an unlimited’/’an illimited’ number of subjects”

As you rightly suspect, there isn’t a strong semantic difference between “unlimited”, “limitless” and “illimited”. There is, however, a sizeable difference in their respective commonnesses (that is, the degree to which each can be considered widely used).

The usual way that a speaker of any language evaluates the commonness of a given word is largely intuitive, and refers instantly to all the words that person has ever heard without a conscious thought. That is to say I could never tell you exactly how many times I’ve heard or seen any one of these words, but I can state with absolute certainty that I’ve encountered “unlimited” much more often than “limitless”, and only seen “illimited” a handful of times while reading very old texts, and while reading legal documents. I don’t believe I’ve ever heard “illimited” said in conversation.

A major difficulty in learning a language without immersion—without years of being inundated with its vocabulary on a daily basis—is that this intuition for a word’s rightness (which extends beyond how common or rare it is) will take a long while of speaking words that sound a little strange to your audience.

The good news is that you are by no means alone in this struggle. It’s something that speakers at every level of every language encounter, especially the more they wish to be understood by more people or by a specific group of people who have a completely different cultural context and/or linguistic background.

For instance, when I lived in the UK, I had to be careful when talking about my pants.

One tool I’ve grown fond of is Google Ngram Viewer. It can be a guide in answering questions like
“how often does ‘illimited’ appear in published works over the years?”
_{(source: static.ow.ly)}

Notice that there are several zeros before the numbers on the left. That means that even at the peak of its popularity, this word was occurring at a proportion of only six times for every ten million words.

This leads nicely to another question, namely
“how often does ‘illimited’ appear compared to ‘limitless’ or ‘unlimited’?”
_{(source: static.ow.ly)}

Because these other two words appear so much more often, the hills and valleys of the previous graph have been flattened by the scale of the display. This is similar to the fact that if you were as much bigger than the Earth as you are bigger than a billiard ball, the Earth would exceed the World Pool-Billiard Association’s specification for the smoothness of a billiard ball.

The upshot of all this is that “unlimited” is the word that more people use in more instances. That means that it is probably understood by more people, and in the broadest sense of our three choices. If you wanted to sound a little more poetic, or felt it captured the specifics in question, you might choose “limitless” instead. If you wanted to convey (or mock) legalistic authority or an archaic tone, you might select “illimited”.