Question

Let \( [P] \) denote the greatest integer \( \leq P \). If \( 0 \leq a \leq 2 \), then the number of integral values of \( a \) such that \( \lim_{x \to a} [x^2] - [x]^2 \) does not exist is:

• A. 3
• B. 2
• C. 1
• D. 0

Hint:

When dealing with the greatest integer function, check for discontinuities at integer points, as the greatest integer function is not continuous at integers.

Answer 1

The Correct Option is B

We are asked to find the number of integral values of \( a \) such that the limit \( \lim_{x \to a} [x^2]
- [x]^2 \) does not exist. Step 1: The expression \( [x^2]
- [x]^2 \) involves the greatest integer function. The limit does not exist when there is a discontinuity in the greatest integer function near \( a \). We examine the behavior of the function near integer points. Step 2: Check the points where the limit does not exist. These occur when \( a \) is an integer, and the values of \( [x^2] \) and \( [x]^2 \) have different behavior at these points. Step 3: The limit does not exist at two specific values of \( a \), giving us the number of integral values of \( a \) as 2. Thus, the number of integral values of \( a \) is 2.