lim ((1 + x)^(1/x) / e)^(1/x)

Logarithm. Let $L(x)=\frac1x\ln\frac{(1+x)^{1/x}}e=\frac1x\left(\frac{\ln(1+x)}x-1\right)$.

Expansion. $\ln(1+x)=x-\frac{x^2}2+\frac{x^3}3+o(x^3)$, so
$$\frac{\ln(1+x)}x-1=-\frac x2+\frac{x^2}3+o(x^2).$$

Limit. Therefore $L(x)=-\frac12+\frac x3+o(x)\to-\frac12$, and by continuity of $\exp$ the limit is
$$e^{-1/2}=\frac1{\sqrt e}.$$

Remark. This also shows that $(1+x)^{1/x}=e\left(1-\frac x2+o(x)\right)$: it approaches $e$ linearly, with slope $-\frac e2$. (Compare with the expansion of $(1+\frac1n)^n$.)