answer:
Let's analyze each statement one by one:
I. The statement says that the limit of f(x) exists.
To determine if the limit exists, we need to check if the left-hand limit and the right-hand limit at x = 1 are equal. Since f(x) is defined as -1 for all x not equal to 1, the left-hand limit and the right-hand limit both equal -1. Therefore, the limit of f(x) as x approaches 1 exists.
II. The statement says that f(1) exists.
Since f(x) is defined as 4 at x = 1, f(1) does exist.
III. The statement says that f is continuous at x = 1.
For a function to be continuous at a point, the limit of the function as x approaches that point must exist, and it must be equal to the value of the function at that point. In this case, the limit of f(x) as x approaches 1 exists and is equal to -1, while f(1) is equal to 4. Therefore, the function is not continuous at x = 1.
Based on our analysis:
- Statement I is true because the limit of f(x) exists.
- Statement II is true because f(1) exists.
- Statement III is false because f is not continuous at x = 1.
Therefore, the correct answer is C) I and II only.
real