# Double Limits!

**Calculus**Level 5

Define the sequence \(\{L_n\}_{n=1}^{\infty}\) as follows:

\[ L_n = \dfrac{1}{\sqrt{n}} \int_{0}^{\infty} \left( 1+ \frac{x}{n} \right)^n e^{-x} \, dx \]

Also, let \(\displaystyle L = \lim_{n \to \infty} L_n\).

- If you come to the conclusion that the sequence diverges (that is \(L\) doesn't exist) enter \(999\).
- If \(L=0\) enter \(0\).
- If \(L\) can be expressed in the form \(\dfrac{a}{b}\), where \(a\) and \(b\) are coprime positive integers, enter \(a+b\).
- If \(L\) is an algebraic real which is not rational, enter \(\lfloor 100L\rfloor\).
- If \(L\) is a transcendental real, enter \(\lfloor 1000L \rfloor\).

**Notation**: \( \lfloor \cdot \rfloor \) denotes the floor function.