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I've encountered the following limit while verifying the Central Limit Theorem for the sum of several exponential random variables: $$\lim_{n\to \infty} \left(1+\frac{x}{\sqrt{n}}\right)^n\exp(-x\sqrt{n})$$ I know the answer, which is $\exp(-x^2/2)$. Can anyone help in computing the limit?

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2 Answers

up vote 4 down vote accepted

Hint: take the logarithm, and use the Maclaurin series for $\log(1+u)$.

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Let $A=\lim_{n\to \infty} \left(1+\frac{x}{\sqrt{n}}\right)^n\exp(-x\sqrt{n})$, therefore $\log A=\lim_{n\to \infty} n\log (1+\frac{x}{\sqrt{n}})-x\sqrt{n}$.Now take $y=1/n$ gives $\log A=\lim_{y\to 0^+} \frac{\log (1+ x\sqrt{y})}{y}-\frac{x}{\sqrt{y}}=\lim_{y\to 0^+} \frac{\log (1+ x\sqrt{y})-x\sqrt{y}}{y}$.Now using L'Hopital's Rule $\log A=-x^2/2\implies A=e^{-x^2/2}$.

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