The answer choice "Caligula was emperor of Rome during the first century; and he was one of the most brutal rulers in history" is incorrect because it contains an ungrammatical use of the semicolon. Two independent clauses can be combined using a comma followed by a conjunction, but not a semicolon followed by a conjunction.

The original text "Caligula was emperor of Rome during the first century, he was one of the most brutal rulers in history" and the answer choices "Caligula was emperor of Rome during the first century, he has been one of the most brutal rulers in history" and "Caligula was emperor of Rome during the first century, he has proven to be one of the most brutal rulers in history" are examples of run-on sentences and are therefore incorrect.

The answer choice "Caligula, one of the most brutal rulers in history, was emperor of Rome during the first century" is the only answer choice that avoids creating a run-on sentence and misusing a semicolon, and is therefore correct.

The two phrases in the sentence have slightly contradictory ideas, with the second one modifying the first. A conjunction is needed to show these differences in the two phrases. The correct answer is the one that uses "but" to separate the two parts.

The sentence is a complex sentence, featuring two complete thoughts put together to form one sentence. To appropriately join the two thoughts, they need to have a semicolon or a comma and a conjunction. "The cold weather can be quite difficult for young children, so companies offer special clothing for infants during the winter months," is the only correct answer.

The second part of the sentence is a dependent clause, one that cannot stand alone as its own complete sentence; therefore, it needs to be appropriately connected to the first portion of the sentence with the correct preposition. Of all the answer choices, only "hotly debated, after new information" does this.

This sentence contains a comma splice, an error in which two independent clauses (sentences that could each stand on their own) are improperly joined by a comma. When joining two independent clauses into a single sentence, one should use either a semicolon or a comma followed by a conjunction. Two of the possible answer choices employ a comma followed by a conjunction, but the answer choice that uses "but" is incorrect, because "but" should join independent clauses that contrast against one another, which the two independent clauses in question do not do. So, the correct answer is the answer choice that uses a comma and "so" to join the two independent clauses, because the relationship between the independent clauses is one of cause and effect, or reason and action taken due to the reason, not one in which the ideas contrast against each other.

The sentence is composed of two independent clauses—two complete sentences, joined to show their connection; however, the two are joined only by a comma, creating an error known as a "comma splice." Independent clauses should be joined by a semi-colon or by a comma followed by a conjunction (a word like "and" or "but"). To correct this sentence, you could either change out the comma for a semi-colon, or insert an appropriate conjunction immediately after the comma.

Let's consider the possible answer choices. The answer choice that makes no changes to the sentence can't be correct, since the sentence does contain an error. "Also" is an adverb, not a conjunction, so the answer choice that uses it as a conjunction cannot be correct. The remaining answer choices properly use "but," "so," and "although" as conjunctions, so finding the right answer becomes a matter of picking the conjunction that matches the logic of the sentence. "But" and "although" are used to indicate that the second sentence contrasts in some way with the first; this wouldn't make sense in our sentences, since they don't contrast with each other, but are related by causation: the first sentence explains the second. "So" is thus the proper conjunction to use, making the correct answer, "She was quite wary of making a scene, so she let the matter fade away with no comment."

The underlined part of the sentence combines two independent clauses: "she realized that she had left her keys at home" and "she had rushed over too quickly." You can combine two independent clauses with a semicolon if they are closely related (such as if one clause causes the other, or if the second clause is explaining or clarifying the first). Only one answer choice correctly uses a semicolon without introducing other errors, and that answer choice is "she had left her keys at home; she had rushed over too quickly." The other options are incorrect because they create run-on sentences or use inconsistent verb tenses.

The answer choice "rocks; for this reason, the period" contains an appropriate use of a semicolon and is correct.

The original text "rocks, it is for this reason that the period" and the answer choice "rocks, for this reason the period" each create a run-on sentence, so neither of these answers is correct.

The answer choice "rocks, which for this reason, the period" contains a misplaced relative clause and is therefore incorrect.

The answer choice "rocks; and for this reason, the period" contains an incorrect use of a semicolon. Sentences can be joined by either a comma followed by a conjunction or by a semicolon, but not by a semicolon followed by a conjunction.

The part of the sentence that precedes the comma and the part of the sentence that comes after the comma are each independent clauses; they could stand as complete sentences by themselves. It is not grammatically correct to join two independent clauses together into a compound sentence by using just a comma; this is a grammatical error known as a "comma splice." Compound sentences may be formed using either a semicolon by itself or a comma followed by a conjunction. The only answer choice that correctly uses one of these methods to fix the sentence's comma splice is "The teacher was a stickler for order; she did not tolerate anyone acting up in her classroom."

The sentence is written as a compound sentence, joining two independent phrases into one sentence to show their relationship. All compound sentences, though, need to be joined either by a semicolon or a comma followed by a conjunction. The only answer choice which employs either of these options is "Television has been a controversial medium since its inception; critics were bemoaning its cultural status as early as the 1940s."