Question

Let \( a = \frac{\sqrt{5}+1}{2} \) and \( b = \frac{\sqrt{5}-1}{2} \). Then, \( \lim_{n \to \infty} \frac{a^n + b^n}{a^n - b^n} \) is ________

• A. is 1
• B. is \(\frac{1}{2}\)
• C. is 0
• D. does not exist

Hint:

When dealing with limits involving powers of numbers, remember that terms with bases less than 1 decay to zero as the exponent increases.

Answer 1

The Correct Option is A

Step 1: Understanding the question.
We are given the values of \(a\) and \(b\), which are the golden ratio \(\phi = \frac{\sqrt{5}+1}{2}\) and its conjugate \(\frac{\sqrt{5}-1}{2}\). As \(n\) tends to infinity, the term \(b^n\) approaches zero because \(b<1\). Thus, we focus on the ratio between the terms involving \(a^n\) and \(b^n\).
Step 2: Analyzing the expression.
\[ \lim_{n \to \infty} \frac{a^n + b^n}{a^n - b^n} = \lim_{n \to \infty} \frac{a^n}{a^n} = 1, \text{ as } b^n \text{ tends to } 0. \] Step 3: Conclusion.
The correct answer is (C) 0, as the ratio will tend to zero due to the behavior of the terms at infinity.