Unit 5: Sequences, Series & Limits

Answer: A — The correct answer is A because as x approaches infinity, the value of 1/x approaches 0. This is because as x gets larger and larger, the fraction 1/x gets smaller and smaller, approaching 0. Option B is incorrect because the limit of f(x) as x approaches infinity is not 1. Option C is incorrect because the limit of f(x) as x approaches infinity is defined, it is 0. Option D is incorrect because the limit of f(x) as x approaches infinity is not infinity, it is 0.

Answer: A — The correct answer is A) The limit of f(x) as x approaches infinity is 0, because as x gets larger and larger, the value of 1/x gets closer and closer to 0. This is a fundamental concept in limits, where the limit of a function as x approaches infinity can be thought of as the behavior of the function as x gets arbitrarily large. Options B, C, and D are incorrect because the function f(x) = 1/x does not approach 1, is not undefined, and does not approach infinity as x approaches infinity.

Answer: A — The correct answer is A because the primary goal of evaluating limits is to determine the behavior of a function as the input values approach a specific point. This involves finding the value that the function approaches as the input values get arbitrarily close to that point. The other options are incorrect because they do not accurately describe the primary goal of evaluating limits. Option B is incorrect because evaluating limits is not primarily about finding maximum or minimum values. Option C is incorrect because evaluating limits does not directly involve graphing the function. Option D is incorrect because evaluating limits is not about solving the function for a specific input value.

Answer: B — The correct answer is B, because the function approaches the same value (4) from both the left and the right side of x = 2. This is evident from the fact that f(2 + δ) = (2 + δ)^2 and f(2 - δ) = (2 - δ)^2 both approach 4 as δ approaches 0. Option A is incorrect because the limit as x approaches 2 depends on the behavior of the function near x = 2, not on whether it is defined for x < 2. Option C is incorrect because we can determine the limit from the given information. Option D is incorrect because the function is continuous at x = 2, as shown by the fact that f(2) = 4 and the limit as x approaches 2 is also 4.

Answer: B — The correct answer is B. The limit of f(x) as x approaches 2 can be determined by examining the function values as x gets arbitrarily close to 2 from both the left and the right. Since the function values approach 4 from both sides, we can conclude that the limit exists and is equal to 4. Option A is incorrect because f(2) is defined and equals 4. Option C is incorrect because the function values do approach a single value from both the left and the right. Option D is incorrect because the function values approach 4, not a value greater than 4.