Limits, Continuity & Differentiability
Limits and Exponential
Grade 12
Question:
<p>Let <span class="math">\(\lim_{n \to \infty} \frac{1}{2} - \frac{1}{2} \left(1 + \frac{1}{n}\right)^{-1} \times \left(1^1 \times 2^2 \times 3^3 \times \ldots \times n^n\right)^{1/n^2} = e\)</span>, where <span class="math">\(p\)</span> and <span class="math">\(q\)</span> are relatively prime positive integers. Find the value of <span class="math">\(|p + q|\)</span>.</p>
Step-by-Step Solution
Key Concept: Apply logarithms to convert the product into a sum, then use properties of limits and the definition of <span class="math">$e$</span>.
<p><strong>Solution:</strong> Use logarithmic properties and Stirling's approximation to evaluate the limit involving the product <span class="math">$1^1 \times 2^2 \times \ldots \times n^n$</span>.</p>
Correct Answer: 5