MSQ

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Okay, so I need to solve this multiple-choice question (MSQ). Let me read it carefully. The question is about a function defined as f(x) = (x^2 - 4)/(x - 2). The options are about the domain of f(x), the limit as x approaches 2, and whether the function is continuous at x = 2.


First, let me recall what the domain of a function is. The domain is all the real numbers for which the function is defined. Since this is a rational function, the main concern is the denominator not being zero. The denominator here is (x - 2), so when x = 2, the denominator becomes zero, which is undefined. Therefore, the domain should be all real numbers except x = 2. Let me check the options. Option A says the domain is all real numbers except x = 2. That seems right. Option B says all real numbers, which would be incorrect because of the division by zero at x = 2. So A is likely correct for the domain part.


Next, the limit as x approaches 2. To find this, I can try plugging in x = 2 directly, but that gives 0/0, which is indeterminate. So I need to simplify the function first. The numerator is x^2 - 4, which factors into (x - 2)(x + 2). So f(x) = [(x - 2)(x + 2)]/(x - 2). As long as x ≠ 2, we can cancel out the (x - 2) terms, leaving f(x) = x + 2. Therefore, the simplified function is x + 2, but with the restriction that x ≠ 2. So the limit as x approaches 2 would be the limit of x + 2 as x approaches 2, which is 2 + 2 = 4. So the limit is 4. Let me check the options. Option C says the limit is 4. That seems correct. Option D says the limit doesn't exist, which is wrong because the limit does exist and is 4.


Now, the third part is about continuity at x = 2. For a function to be continuous at a point, three conditions must be met: 1) f(a) is defined, 2) the limit as x approaches a exists, and 3) the limit equals f(a). In this case, a is 2. The function f(x) is not defined at x = 2 because the original function has a denominator of zero there. Therefore, even though the limit as x approaches 2 is 4, the function isn't defined at x = 2, so it can't be continuous there. Option E says the function is continuous at x = 2, which is false. Option F says it's not continuous, which is true. Wait, but the options might not be labeled E and F here. Let me check the original problem again.


Wait, the user mentioned options A to F, but in the initial problem statement, the options are listed as A, B, C, D, E, F. Let me confirm. The user wrote:


"A) The domain of f(x) is all real numbers except x = 2.
B) The domain of f(x) is all real numbers.
C) The limit of f(x) as x approaches 2 is 4.
D) The limit of f(x) as x approaches 2 does not exist.
E) The function f(x) is continuous at x = 2.
F) The function f(x) is not continuous at x = 2."


So, based on that, E is false and F is true. Therefore, for the continuity part, F is correct.


Putting it all together: A is correct for the domain, C is correct for the limit, and F is correct for continuity. However, the problem might be a multiple-select question where more than one option is correct. Let me check the original question again. The user wrote "MSQ" which could mean multiple-select question. So the correct answers would be A, C, and F.


Wait, but sometimes in such questions, each part is a separate option. Let me make sure. The way the options are presented, each option is a separate statement. So the question might be asking which of the statements A-F are correct. So all three correct statements (A, C, F) should be selected. However, depending on the exam's format, sometimes only one option is correct. Wait, the user hasn't specified whether it's multiple correct answers or just one. But given that it's an MSQ (multiple-select question), likely more than one is correct.


But let me double-check. The function f(x) simplifies to x + 2 except at x = 2. So the domain is indeed all real numbers except 2 (A). The limit as x approaches 2 is 4 (C). The function is not continuous at x = 2 because it's not defined there (F). Therefore, the correct options are A, C, F.


But let me check if there's any trick here. For example, sometimes people might argue that if you redefine the function at x = 2 to be 4, then it becomes continuous. But the original function is not defined there, so unless there's a piecewise definition, which there isn't, the function remains discontinuous at x = 2. So F is correct.


Therefore, the correct answers are A, C, and F. However, depending on how the options are structured, maybe the question expects only one correct answer. Wait, the user hasn't specified, but given the options are labeled A-F, and the question is MSQ, which usually allows multiple correct answers. So the answer should be A, C, F.




The function ( f(x) = \frac{x^2 - 4}{x - 2} ) simplifies to ( f(x) = x + 2 ) for all ( x \neq 2 ). Here's the analysis:



Domain: The denominator ( x - 2 ) is zero at ( x = 2 ), so the domain excludes ( x = 2 ).

Correct Option: A



Limit as ( x \to 2 ): After simplification, ( \lim_{x \to 2} f(x) = \lim_{x \to 2} (x + 2) = 4 ).



Correct Option: C



Continuity at ( x = 2 ): ( f(x) ) is undefined at ( x = 2 ), violating the first condition for continuity.

Correct Option: F




Final Answer:

\boxed{A}, \boxed{C}, \boxed{F}