<s>[INST] Calculate $\lim{a_{n}}$ for $a_{n}=((1+\frac{1}{n^{2}})^{n^{2}}(1-\frac{1}{n})^{n}(1+\frac{1}{n}))^{n}$ $$a_{n}=((1+\frac{1}{n^{2}})^{n^{2}}(1-\frac{1}{n})^{n}(1+\frac{1}{n}))^{n}=(e^{\ln((1+\frac{1}{n^{2}})^{n^{2}})+\ln((1-\frac{1}{n})^{n})+\ln((1+\frac{1}{n})^{n})})^{n}$$Then I can use Taylor's theorem - because function $f(x)=e^{x}$ is monotonic I can examine $\ln((1+\frac{1}{n^{2}})^{n^{2}})+\ln((1-\frac{1}{n})^{n})+\ln((1+\frac{1}{n})^{n})$. However I think it is too time-consuming and there is probably a faster way.Have you some ideas?
 [/INST] You know $$\left(1 + \frac{1}{n}\right)^n \to e$$ as $n \to \infty$. As for
$$\left(1 + \frac{1}{n^2}\right)^{n^3} \left(1 - \frac{1}{n}\right)^{n^2}$$
take logarithms and use $\log(1 + x) = x - x^2/2 + \dots$ as $x \to 0$ to get 
$$ n^3 \log\left(1+\frac{1}{n^2}\right) + n^2 \log\left(1 - \frac{1}{n}\right)= n - \dots - n - \frac{1}{2} - \dots \to -\frac{1}{2}$$ as $n \to \infty$. 
Thus the limit is $$e^{-1/2} e = \sqrt{e}.$$
</s>