<s>[INST] Proving $\lim_{x\rightarrow 1}x^{\frac{1}{n-x-x^2-x^3-\ldots -x^n}}=\frac{1}{\sqrt[k]{e}}$ Proving 

$$\lim_{x\rightarrow 1}x^{\left(\frac{1}{n-x-x^2-x^3\cdots -x^n}\right)}=\frac{1}{\sqrt[k]{e}}$$
when $$k=1+2+3+\ldots+n$$

 [/INST] Hint: $$x^{\left(\frac{1}{n-x-x^2-x^3\cdots -x^n}\right)}=\exp\left(\ln x^{\left(\frac{1}{n-x-x^2-x^3\cdots -x^n}\right)}\right)=\exp\left(\frac{\ln x}{n-x-x^2-\ldots-x^n}\right)$$ and now due to continuity of $\exp$ it suffices to show with L' Hopitals rule that $$\frac{\ln x}{n-x-x^2-\ldots-x^n} \to -\frac{1}{1+2+3+\ldots+n}=-\frac{1}{k}$$ as $x\to 1$.

Indeed $$\begin{align*}\lim_{x \to 1} \frac{\ln x}{n-x-x^2-\ldots-x^n}&\overset{\frac{0}{0}}=\lim_{x\to 1}\frac{(\ln x)'}{(n-x-x^2-\ldots-x^n)'}\\&=\lim_{x \to 1}\frac{\frac1x}{-1-2x-3x^2-\ldots-nx^{n-1}}\\[0.2cm]&=\frac{1}{-1-2-3-\ldots-n}=-\frac{1}{1+2+3+\ldots+n}=-\frac{1}{k}\end{align*}$$
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