Limits, Continuity & Differentiability
General
Grade None
Question:
<p>Which of the following limits equal 1/2?</p>
lim n\to \infty \sum_{k=1}^{n} \frac{1}{(2k -1)(2k + 1)}
lim x\to 1 \frac{ \sqrt{3x^2} }{2x + 1} - \frac{(2x -1)(3x^2 + x + 2)}{4x^2}
lim n\to \infty \frac{1}{n^2} (1 + 2 + \cdot \cdot \cdot + n)
lim n\to \infty \frac{(n + 2)! + (n + 1)!}{(n + 2)! - (n + 1)!}
Step-by-Step Solution
Key Concept: General
<p>• (A) Telescope: 1</p> 2 P( 1 2k-1 - 1 2k+1) = 1 2(1 -0) = 1/2. • (B) At x = 1, value is 3/3 -(1)(6)/4 = 1 -1.5 = -0.5. • (C) n(n+1) 2n2 \to 1/2. • (D) (n+2)+1 (n+2)-1 \to 1.
Correct Answer: (A, C)