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Question #40577

Difficulty: 8/10 | 🟢 Free

The Question

In a continuous-time model of a combinatorial process, the sample space is partitioned into valid arrangements and total selections. The rate of generating valid arrangements is given by $a(t) = t(5 + 2\sin(t^2))$ , while the rate of generating total selections is $s(t) = t(7 + 2\sin(t^2))$ .

The cumulative measure of valid arrangements and total selections up to time $x$ are $A(x) = \int_0^x a(t) dt$ and $S(x) = \int_0^x s(t) dt$ , respectively.

The asymptotic probability of an arrangement being valid is defined as $P = \lim_{x \to \infty} \frac{A(x)}{S(x)}$ .

Find the value of $P$ .

Options

A.
$\frac{5}{7}$
B.
$\frac{2}{7}$
C.
0
D.
\text{Does not exist}

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Selection vs Arrangement|Sample Space Partitioning|L'Hopital's Pitfall

Why This Question Is Tricky

Common Mistakes

This question tests the student's understanding of the conditions required for L'Hopital's Rule, specifically the converse pitfall.

When evaluating the asymptotic probability $P = \lim_{x \to \infty} \frac{A(x)}{S(x)}$ , the form is $\infty/\infty$ .

A careless application of L'Hopital's Rule (differentiating the integrals via the Fundamental Theorem of Calculus) yields a limit that oscillates and does not exist.

Students fall into the trap of concluding the original limit also does not exist.

However, L'Hopital's Rule only implies that if the limit of the derivatives exists, it equals the limit of the functions.

By manually integrating the rates, the bounded oscillating term $\cos(x^2)$ is overpowered by the $x^2$ term, revealing the true probability of $5/7$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Permutations and Combinations - Class 11
Concept	Selection vs Arrangement
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free