Tuesday, 17 January 2017

real analysis - Show that limnrightarrowinftyfracsinn(fracxsqrtn)left(fracxsqrtnright)n=efracx26




I am wondering about a limit that wolframalpha got me and that you can find here wolframalpha



It says that lim



Does anybody know if there is a "easy" way to get this?


Answer



As
1 - \frac{\sin t}t \sim_{t\to 0} \frac{t^2}{3!} \\ \log (1+\epsilon) \sim_{\epsilon\to 0} \epsilon



you have
\frac{\sin^{n}(\frac{x}{\sqrt{n}})}{\left(\frac{x}{\sqrt{n}} \right)^n} = \exp \left[ n\log \frac{\sin \frac x{\sqrt n}}{\frac x{\sqrt n}} \right] \to \exp \left[ n\left( -\frac 16 \left(\frac x{\sqrt n}\right)^2 \right) \right] = \exp\left( -\frac{x^2}6\right)


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