The following expression, where the process continues indefinitely, is called a continued fraction. Complete parts (a) through (e). 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.) B. The limit does not exist.

To find the limit of the continued fraction as \( n \rightarrow \infty \), we can set up the equation based on the assumption that the limit exists, denoted as \( L \). If the continued fraction is structured such that \( a_n = a + \frac{1}{L} \), then we can rearrange this into the equation: \[ L = a + \frac{1}{L} \] Multiplying both sides by \( L \) gives us: \[ L^2 = aL + 1 \] Rearranging this as a quadratic equation leads to: \[ L^2 - aL - 1 = 0 \] Using the quadratic formula, we find: \[ L = \frac{a \pm \sqrt{a^2 + 4}}{2} \] So, the correct choice is A. The limit is \( \frac{a + \sqrt{a^2 + 4}}{2} \). This means you would fill the answer box for choice A with \( \frac{a + \sqrt{a^2 + 4}}{2} \) and circle it with pride! Now go ahead and solve the entire expression with confidence!