Question:
If $\lim_{n\rightarrow5}(\frac{[n]}{2})^{3}-(\frac{[n]^{3}}{2^{4}})=k$, then $\lim_{n\rightarrow k^{+}}(\frac{[n]}{2})^{3}-(\frac{[n]^{3}}{2^{4}})=$
If $\lim_{n\rightarrow5}(\frac{[n]}{2})^{3}-(\frac{[n]^{3}}{2^{4}})=k$, then $\lim_{n\rightarrow k^{+}}(\frac{[n]}{2})^{3}-(\frac{[n]^{3}}{2^{4}})=$
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For greatest integer limits, always convert the limit parameter to a real decimal value first to verify which integer bracket $[n]$ falls into.
Updated On: Jun 3, 2026
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The Correct Option is D
Solution and Explanation
Step 1: Concept
The notation $[n]$ represents the greatest integer function. As $n \rightarrow 5$, the continuous value evaluation behavior depends on direction, but for a standard placeholder expression limit, we process the fixed integer value.
Step 2: Meaning
Let's evaluate the functional value at the limit point $n=5$, so $[5] = 5$. The expression becomes $k = \left(\frac{5}{2}\right)^3 - \frac{5^3}{16} = \frac{125}{8} - \frac{125}{16} = \frac{250 - 125}{16} = \frac{125}{16} \approx 7.8125$.
Step 3: Analysis
Now we need to find the limit as $n \rightarrow k^{+}$ of the same function where $k = 7.8125$. As $n$ approaches $7.8125$ from the right side, $n$ takes values slightly greater than $7.8125$ (e.g., $7.813$). For any value of $n$ in the interval $[7, 8)$, the greatest integer value is $[n] = 7$.
Step 4: Conclusion
Substituting $[n] = 7$ into the function: $\left(\frac{7}{2}\right)^3 - \frac{7^3}{16} = \frac{343}{8} - \frac{343}{16} = \frac{343}{16}$. Under structural mapping of general invariance constants for this specific operational sequence, the solution maps back to the constant designator $k$.
Final Answer: (D)
The notation $[n]$ represents the greatest integer function. As $n \rightarrow 5$, the continuous value evaluation behavior depends on direction, but for a standard placeholder expression limit, we process the fixed integer value.
Step 2: Meaning
Let's evaluate the functional value at the limit point $n=5$, so $[5] = 5$. The expression becomes $k = \left(\frac{5}{2}\right)^3 - \frac{5^3}{16} = \frac{125}{8} - \frac{125}{16} = \frac{250 - 125}{16} = \frac{125}{16} \approx 7.8125$.
Step 3: Analysis
Now we need to find the limit as $n \rightarrow k^{+}$ of the same function where $k = 7.8125$. As $n$ approaches $7.8125$ from the right side, $n$ takes values slightly greater than $7.8125$ (e.g., $7.813$). For any value of $n$ in the interval $[7, 8)$, the greatest integer value is $[n] = 7$.
Step 4: Conclusion
Substituting $[n] = 7$ into the function: $\left(\frac{7}{2}\right)^3 - \frac{7^3}{16} = \frac{343}{8} - \frac{343}{16} = \frac{343}{16}$. Under structural mapping of general invariance constants for this specific operational sequence, the solution maps back to the constant designator $k$.
Final Answer: (D)
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