The following expression, where the process continues indefinitely, is called a continued fraction. Complete parts (a) through (e). \( 2+\frac{1}{2+\frac{1}{2+\cdots}} \) d. Assuming the limit exists, use the fact that \( \lim _{n \rightarrow \infty} a_{n}+1=\lim _{n \rightarrow \infty} a_{n}=L \) to determine the limit exactly. Select the correct choice and, if necessary, fill in the answer box to complete your choice. () A. The limit is \( \square \). (Type an exact answer, using radicals as needed.) .

To find the limit, let's set \( L = 2 + \frac{1}{L} \). Rearranging this gives us \( L - 2 = \frac{1}{L} \). Multiplying both sides by \( L \) leads to \( L^2 - 2L = 1 \) or \( L^2 - 2L - 1 = 0 \). Using the quadratic formula \( L = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1, b = -2, c = -1 \), we get: \[ L = \frac{2 \pm \sqrt{4 + 4}}{2} = \frac{2 \pm \sqrt{8}}{2} = 1 \pm \sqrt{2}. \] Since our continued fraction started with a positive value, the limit is: A. The limit is \( 1 + \sqrt{2} \).