# 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.

Inspiration

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