Unit 1: Limits and Continuity

Answer: A — The correct answer is B. Since the limit of f(x) as x approaches 2 from the right is positive infinity, this means that as x gets arbitrarily close to 2 from the right, f(x) gets arbitrarily large. However, this does not provide any information about the behavior of the function as x approaches 2 from the left, so it may approach negative infinity. Options A and D are incorrect because if the limit of f(x) as x approaches 2 from the right is positive infinity, then the function cannot be continuous at x = 2 and it cannot have a local maximum at x = 2. Option C is incorrect because the limit of f(x) as x approaches 2 from the left may be positive infinity, negative infinity, or a finite number.

Answer: B — The correct answer is A) 4 because the function f(x) = (x^2 - 4) / (x - 2) can be simplified to f(x) = (x + 2)(x - 2) / (x - 2), which further simplifies to f(x) = x + 2 for x ≠ 2. As x approaches 2, the limit of f(x) approaches 2 + 2 = 4. The other options are incorrect because option B) 0 is not the limit, option C) 2 is the value that x approaches but not the limit of the function, and option D) The limit does not exist is incorrect because the limit can be determined by simplifying the function.

Answer: B — The correct answer is B because as x approaches 0 from the right, the values of 1/x increase without bound, approaching positive infinity. This is a fundamental concept in limits, where the behavior of the function as it approaches a certain point is analyzed. Option A is incorrect because the limit can still exist even if the function is not defined at the point. Option C is incorrect because the function values actually increase without bound, not decrease. Option D is incorrect because the function is not continuous at x = 0 and the limit is not 1.

Answer: C — The correct answer is D) The limit is 4. Although the function f(x) is not defined at x = 2, the limit as x approaches 2 can still be found. By factoring the numerator, we get f(x) = ((x + 2)(x - 2)) / (x - 2). Cancelling out the (x - 2) terms, we are left with f(x) = x + 2 for x ≠ 2. Thus, the limit of f(x) as x approaches 2 is 4. The other options are incorrect because they do not take into account the simplification of the function. Option A is incorrect because the limit can exist even if the function is not defined at a point. Option B is incorrect because the numerator is not 0 when x approaches 2. Option C is incorrect because the denominator being 0 does not necessarily mean the limit is undefined.

Answer: D — The correct answer is D because the limit of a sum is the sum of the limits. This is a fundamental property of limits. Options A and B are incorrect because they describe the limit of a difference and the limit of a product, respectively. Option C is incorrect because if the limits of f(x) and g(x) as x approaches a exist, then the limit of f(x) + g(x) as x approaches a will also exist.