Question

Let \([x]\) denote the greatest integer less than or equal to \(x\). Then \[ \lim_{x \to 2^+} \left( \frac{[x]^3}{3} - \left[ \frac{x^3}{3} \right] \right) \]

• A. \( 0 \)
• B. \( \frac{8}{3} \)
• C. \( \frac{64}{27} \)
• D. \( \frac{1}{3} \)

Hint:

When handling limits involving the greatest integer function, use small increments from the direction of approach and evaluate step-by-step with numerical substitution.

Answer 1

The Correct Option is B

Step 1: We are asked to evaluate the right-hand limit: \[ \lim_{x \to 2^+} \left( \frac{[x]^3}{3} - \left[ \frac{x^3}{3} \right] \right) \] Step 2: When \( x \to 2^+ \), we are approaching 2 from the right (i.e., values like 2.01, 2.001, etc.). So: \[ [x] = 2
\text{(since greatest integer less than or equal to \( x \) is 2)} \Rightarrow [x]^3 = 8 \Rightarrow \frac{[x]^3}{3} = \frac{8}{3} \] Step 3: Now consider \( \left[ \frac{x^3}{3} \right] \) As \( x \to 2^+ \), we choose values slightly greater than 2: Let \( x = 2 + h \), where \( h \to 0^+ \) Then: \[ x^3 = (2 + h)^3 = 8 + 12h + 6h^2 + h^3 \Rightarrow \frac{x^3}{3} = \frac{8}{3} + 4h + 2h^2 + \frac{h^3}{3} \] Since \( h \to 0^+ \), this expression is slightly more than \( \frac{8}{3} \), but still less than 3 (since \( \frac{8}{3} \approx 2.666\ldots \)). So: \[ \left[ \frac{x^3}{3} \right] = 2 \] Step 4: Substitute back: \[ \frac{[x]^3}{3} - \left[ \frac{x^3}{3} \right] = \frac{8}{3} - 2 = \frac{2}{3} \] Correction: Wait! This gives \( \frac{2}{3} \), not \( \frac{8}{3} \). But the answer marked in the image is \( \frac{8}{3} \), which suggests a misinterpretation. Let's reevaluate: Actually, mistake caught: When \( x \to 2^+ \), the floor of \( \frac{x^3}{3} \) is: Try specific values: - If \( x = 2.01 \), then: \[ x^3 = 8.1206 \Rightarrow \frac{x^3}{3} = 2.7068 \Rightarrow \left[ \frac{x^3}{3} \right] = 2 \] So: \[ \frac{[x]^3}{3} = \frac{8}{3},
\left[ \frac{x^3}{3} \right] = 2 \Rightarrow \frac{8}{3} - 2 = \boxed{\frac{2}{3}} \] Thus, the correct final value is: \[ \lim_{x \to 2^+} \left( \frac{[x]^3}{3} - \left[ \frac{x^3}{3} \right] \right) = \frac{2}{3} \] However, the image marks \( \frac{8}{3} \) as the correct option, which appears to be a mistake. % Final Answer Final Value: \( \frac{2}{3} \)