Since the underlined portion of the sentence contains consecutive adjectives, check whether each adjective is used correctly. In the phrase a seeming limitless number, the adjective limitless is used correctly since it modifies the noun number, but since the adjective seeming modifies the adjective limitless, the correct construction is the adverb form seemingly (since an adverb, not another adjective, correctly modifies an adjective). This is an idiom error, so eliminate choice A and look for any obvious repeaters. Choices B and C are obvious repeaters since they also use the phrase seeming limitless, so eliminate choices B and C. Now evaluate the remaining answer choices individually, looking for reasons to eliminate each.
Choice D fixes the idiom error by correctly using the adverb seemingly to modify the adjective limitless, but incorrectly uses the plural pronoun all to refer to the singular noun number, and uses the construction arising out of, an incorrect form of the idiom arising from, so eliminate choice D. Choice E fixes the idiom error by correctly using the adverb seemingly to modify the adjective limitless, and introduces no new errors, so keep choice E.
Alternatively, if it is difficult to spot the grammar rule the sentence is testing, another strategy is to identify an error from the answers by looking for either a 2/3 split or differences among the answers. Because three of the answer choices use the phrase seeming limitless and two answer choices use the phrase seemingly limitless, this difference is an indication to look for idiom errors.
Choice A: No. Seemingly limitless is incorrectly written as seeming limitless. Idiom.
Choice B: No. Seemingly limitless is incorrectly written as seeming limitless. Idiom.
Choice C: No. Seeminglylimitless is incorrectly written as seeming limitless. Idiom.
Choice D: No. The plural pronoun all incorrectly refers to the singular noun number. Pronoun agreement.
Choice E: Correct
The correct answer is choice E.