The following expression, where the process continues indefinitely, is called a continued fraction. Complete parts (a) through (e). \[ 2+\frac{1}{2+\frac{1}{2+\frac{1}{2+\cdots}}} \] c. Using computation and/or graphing, estimate the limit of the sequence, if it exists. Select the correct choice and, if necessary, fill in the answer box to complete your choice. A. The limit appears to be approximately \( \square \) . (Type an integer or a decimal rounded to three decimal places as needed.) B. The limit does not appear to exist.

To analyze the expression, let's denote the continued fraction as \( x \): \[ x = 2 + \frac{1}{x} \] To solve for \( x \), we can multiply both sides by \( x \) to eliminate the fraction: \[ x^2 = 2x + 1 \] Rearranging gives us the quadratic equation: \[ x^2 - 2x - 1 = 0 \] Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1, b = -2, c = -1 \): \[ x = \frac{2 \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1} = \frac{2 \pm \sqrt{4 + 4}}{2} = \frac{2 \pm \sqrt{8}}{2} = \frac{2 \pm 2\sqrt{2}}{2} = 1 \pm \sqrt{2} \] Since the continued fraction should yield a positive result, we take: \[ x = 1 + \sqrt{2} \] Calculating the approximate value gives: \[ \sqrt{2} \approx 1.414 \implies x \approx 1 + 1.414 = 2.414 \] Finally, rounding to three decimal places, we find that the limit appears to be approximately \( 2.414 \). So the answer is: A. The limit appears to be approximately \( 2.414 \).