Question

If \([x]\) denotes the greatest integer less than or equal to \(x\), then \[ \lim_{x\to 0^-}\frac{\sin([x])}{[x]} \] is equal to ________.

• A. 1
• B. $\sin 1$
• C. -1
• D. 0
• E. $-\sin 1$

Hint:

For $x \to 0^{-}$, the floor function $[x]$ is a constant value of $-1$.

Answer 1

The Correct Option is B

Step 1: Concept
For the left-hand limit as $x \to 0^{-}$, the value of $x$ is slightly less than 0 (e.g., -0.1).

Step 2: Meaning
The greatest integer $[x]$ for any value in the interval $(-1, 0)$ is $-1$.

Step 3: Analysis
Substitute $[x] = -1$ into the limit expression: $\frac{\sin(-1)}{-1}$.

Step 4: Conclusion
Since $\sin(-\theta) = -\sin\theta$, we get $\frac{-\sin 1}{-1} = \sin 1$. Final Answer: (B)