Question

For each statement, write the null and alternative hypotheses. State which hypothesis represents the claim.

A. Null: The limit exists; Alternative: The limit does not exist. (Claim: The limit exists)

B. Null: The limit exists; Alternative: The limit does not exist. (Claim: The limit does not exist)

C. Null: The limit approaches 5; Alternative: The limit approaches 2. (Claim: The limit approaches 5)

D. Null: The limit approaches 0; Alternative: The limit approaches [infinity]. (Claim: The limit approaches 0)

Answer 1

A. Null: The limit exists; Alternative: The limit does not exist. (Claim: The limit exists)

B. Null: The limit exists; Alternative: The limit does not exist. (Claim: The limit does not exist)

C. Null: The limit approaches 5; Alternative: The limit approaches 2. (Claim: The limit approaches 5)

D. Null: The limit approaches 0; Alternative: The limit approaches [infinity]. (Claim: The limit approaches 0)

Step-by-step explanation:

In hypothesis testing regarding limits in mathematics, the null hypothesis (H0) typically represents the claim or assumption to be tested. For statement A, the claim is that the limit exists, making the null hypothesis state that the limit does exist. Conversely, for statement B, the claim is that the limit does not exist, thereby the null hypothesis asserts that the limit does exist.

Regarding statement C, the claim is that the limit approaches 5, hence the null hypothesis sets the limit as approaching 5. Finally, for statement D, where the claim is that the limit approaches 0, the null hypothesis states that the limit does approach 0. These hypotheses serve as the baseline for hypothesis testing, allowing mathematicians to perform tests to validate or refute the claims made about the limits of functions or sequences.