$\displaystyle \lim_{n\to\infty}\int_{0}^{\sqrt{n}}\left(1-\frac{\left x^2\right}{n}\right )^{n}dx$ i couldn't tackle it thanks for now.
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Do you know the dominated convergence theorem?
It can be useful the substitution $\displaystyle t=x/\sqrt{n}$, so $\displaystyle \int_{0}^{\sqrt{n}}(1-x^2/n)^n\;dx=\sqrt{n}\int_{0}^{1}(1-t^2)^n\;dt$ .
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