Question

If \([\,]\) denotes the greatest integer function, then \[ \lim_{x\to \frac{\pi}{2}^{+}} \frac{[\sin x]-[\cos x]+1}{2} = \]

• A. \(0\)
• B. \(-\frac{1}{2}\)
• C. \(\frac{1}{2}\)
• D. \(1\)

Hint:

For greatest integer function problems, first determine the interval in which the function value lies. Near \(x=\frac{\pi}{2}^{+}\), \[ 0<\sin x<1 \Rightarrow [\sin x]=0, \] and \[ -1<\cos x<0 \Rightarrow [\cos x]=-1. \]

Answer 1

The Correct Option is D

Step 1: Analyze the greatest integer of sin x as x → (π/2)⁺.
Just above π/2, sin x is slightly less than 1 but still positive. Hence [sin x] = 0.

Step 2: Analyze the greatest integer of cos x as x → (π/2)⁺.
Just above π/2, cos x becomes a small negative number, so [cos x] = -1.

Step 3: Substitute into the given expression.
([sin x] - [cos x] + 1)/2 = (0 - (-1) + 1)/2 = 2/2 = 1.

Step 4: Conclude the limit.
Since the expression is constant in a right neighbourhood of π/2, the limit is 1.

Step 5: Final conclusion.
The limit equals 1.