Limits, Continuity & Differentiability
Standard Limits
Grade 12
Question:
<p>\(\lim_{x \to \infty} C_n^m x^m \left(1 + \frac{1}{n}\right)^{nx} \left(1 + \frac{1}{x}\right)^m\) equals to</p>
<p>(A) \(\frac{m! \cdot e}{x}\)</p>
<p>(B) \(\frac{m! \cdot e}{x}\)</p>
<p>(C) \(e^0\)</p>
<p>(D) \(\frac{m}{2}\)
Step-by-Step Solution
Key Concept: Use standard limit forms: $\lim_{n \to \infty}(1 + 1/n)^n = e$ and analyze the behavior as both variables approach infinity.
<p><strong>Step 1:</strong> Recognize that $\left(1 + \frac{1}{n}\right)^{nx} \to e^x$ as $n \to \infty$.</p><p><strong>Step 2:</strong> $\left(1 + \frac{1}{x}\right)^m \to 1$ as $x \to \infty$.</p><p><strong>Step 3:</strong> The limit becomes $C_n^m \cdot m! \cdot e$ in appropriate limit context.</p>
Correct Answer: A