Limits, Continuity & Differentiability
Limits involving floor functions
Grade 12
Question:
<p>The value of $\lim_{n \to \infty} \frac{1}{n}\left([1^2 x^7 + 1^2] + [2^2 x^7 + 2^2] + \ldots + [n^2 x^7 + n^2]\right)$ (where $[\cdot]$ denotes the greatest integer function) is</p>
<p>(a) $\frac{1}{3}$</p>
<p>(b) $x + \frac{1}{3}$</p>
<p>(c) $\frac{x}{3} + \frac{1}{3}$</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: Apply summation formulas and properties of the greatest integer function to evaluate the limit as $n \to \infty$.
<p>This problem requires evaluating a limit involving the greatest integer function applied to a sum of expressions. The dominant term for large $n$ is $[n^2 x^7]$, and using properties of the floor function and summation formulas, the limit evaluates to $\frac{1}{3}$.</p>
Correct Answer: A