Question

If [.] denotes the greatest integer function.

Statement 1: $\left[ \lim_{x \to 0} \frac{sin x}{x} = 1 \right]$

Statement 2: $\left[ \lim_{x \to 0} \frac{sin x}{x} \right] = 1 \alpha \rightarrow \lim_{x \to 0} \left[ \frac{sin x}{x} \right] = 0$

• A. Statement-1 is True, Statement-2 is True and Statement-2 is a correct explanation for Statement-1.
• B. Statement-1 is True, Statement-2 is True and Statement-2 is NOT a correct explanation for Statement-1.
• C. Statement-1 is True, Statement-2 is False.
• D. Statement-1 is False, Statement-2 is True.

Answer 1

Answer: (B)

Statement-1 is True, Statement-2 is True and Statement-2 is NOT a correct explanation for Statement-1.

Answer 2

Statement 1 Analysis: We know the standard limit:

$$\lim_{x \to 0} \frac{\sin x}{x} = 1$$

Substituting this value into the greatest integer function:

$$\left[ 1 \right] = 1$$

So, Statement 1 simplifies to $1 = 1$, which is True.

Statement 2 Analysis: This statement comprises two parts.

Part 1: $\left[ \lim_{x \to 0} \frac{\sin x}{x} \right] = 1$ As shown in Statement 1's analysis, $\lim_{x \to 0} \frac{\sin x}{x} = 1$. So, $\left[ 1 \right] = 1$. This part is True.

Part 2: $\lim_{x \to 0} \left[ \frac{\sin x}{x} \right] = 0$ For $x$ very close to 0 but not equal to 0, we know that $0 < \frac{\sin x}{x} < 1$. Since $\frac{\sin x}{x}$ approaches 1 from values less than 1, the greatest integer of $\frac{\sin x}{x}$ will be 0:

$$\left[ \frac{\sin x}{x} \right] = 0 \quad \text{for } x \text{ close to } 0, x \neq 0$$

Therefore,

$$\lim_{x \to 0} \left[ \frac{\sin x}{x} \right] = 0$$

This part is True.

Assuming '$\alpha \rightarrow$' signifies logical implication ("If P, then Q"), then Statement 2 is "If (True), then (True)", which makes the entire Statement 2 True.

Relationship between Statement 1 and Statement 2: Statement 2 highlights a key distinction in limits involving the greatest integer function: $\left[ \lim_{x \to a} f(x) \right]$ is not necessarily equal to $\lim_{x \to a} \left[ f(x) \right]$. While Statement 2 provides valuable context by showing this difference, it does not directly explain why Statement 1 is true. Thus, Statement 2 is NOT a correct explanation for Statement 1.