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Question #40937

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \lim_{n \to \infty} \frac{1}{n^4} \sum_{r=1}^{n} \left( \left[ r^3 x \right] + r \left[ r x \right]^2 + r^2 \left[ r x \right] \right)$ , where $[ \cdot ]$ denotes the greatest integer function. If $x > 0$ and $f(x) = 6$ , find the value of $x$ .

Options

A.
2
B.
4
C.
6
D.
8

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Squeeze Theorem (Sandwich Theorem)|Sum of Powers of Natural Numbers

Why This Question Is Tricky

Common Mistakes

This question tests the rigorous application of the Squeeze (Sandwich) Theorem to evaluate limits of sums involving the Greatest Integer Function (GIF).

A common trap is attempting to convert the sum directly into a definite integral using Riemann sums (e.g., $\lim \frac{1}{n} \sum f(r/n) = \int f(x) dx$ ).

However, because the arguments of the GIFs (like $r^3 x$ ) scale with $n$ and go to infinity, the standard Riemann sum form is violated.

Students who blindly replace the sum with an integral of a GIF will encounter undefined or incorrect expressions.

The mathematically sound approach forces the use of the bounding inequality $y - 1 < [y] \le y$ to 'flatten' the discontinuities, allowing the Squeeze Theorem to evaluate the limit using standard sums of powers of natural numbers.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Limits and Derivatives - Class 11
Concept	Squeeze Theorem (Sandwich Theorem)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free