Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
up vote 4 down vote accepted

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

share|cite|improve this answer

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 $\log A=-x^2/2\implies A=e^{-x^2/2}$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.