e. Assuming the limit exists, use the same ideas to determine the value of \( a+\frac{b}{\left[a+\frac{b}{a+\frac{b}{a+\cdots}}\right]} \), where \( a \) and \( b \) are positive real numbers. 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 solve for the value of \( L = a + \frac{b}{L} \), we can start by recognizing that this expression leads to a quadratic equation. Multiplying both sides by \( L \) gives us: \[ L^2 = aL + b \] Rearranging this into standard quadratic form: \[ L^2 - aL - b = 0 \] We can apply the quadratic formula \( L = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \): \[ L = \frac{a \pm \sqrt{a^2 + 4b}}{2} \] Since \( a \) and \( b \) are positive, we only take the positive root (to maintain \( L \) as a positive real number). Thus, we find: \[ L = \frac{a + \sqrt{a^2 + 4b}}{2} \] A. The limit is \( \frac{a + \sqrt{a^2 + 4b}}{2} \).