Question

The value of \(\lim_{x \to \infty} \frac{x}{x + \frac{\sqrt[3]{x}}{x + \frac{\sqrt[3]{x}}{x + \dots}}}\) is

• A. 0
• B. 1
• C. 2
• D. 1/2

Hint:

Set up a recursive equation \(y = f(y)\) and take the limit.

Answer 1

The Correct Option is B

Step 1: Formula / Definition}
\[ y = \frac{x}{x + \frac{\sqrt[3]{x}}{x + \dots}} \Rightarrow y = \frac{x}{x + \frac{\sqrt[3]{x}}{y}} \]
Step 2: Calculation / Simplification}
\[ y = \frac{x}{x + \frac{x^{1/3}}{y}} = \frac{xy}{xy + x^{1/3}} \]
\(xy^2 + x^{1/3}y - xy = 0 \Rightarrow y^2 + x^{-2/3}y - y = 0\)
As \(x \to \infty\), \(x^{-2/3} \to 0 \Rightarrow y^2 - y = 0\)
\(y(y-1) = 0 \Rightarrow y = 1\) (since \(y>0\))
Step 3: Final Answer
\[ 1 \]