\(\lim _{x \rightarrow \infty} x^3\left\{\sqrt{x^2+\sqrt{1+x^4}}-x \sqrt{2}\right\}=\)

\(\begin{aligned} & \lim _{x \rightarrow \infty} x^3\left(\sqrt{x^2+\sqrt{1+x^4}}-x \sqrt{2}\right) \\ & =\lim _{x \rightarrow \infty} \frac{x^3\left(x^2+\sqrt{1+x^4}-2 x^2\right)}{\sqrt{x^2+\sqrt{1+x^4}}+x \sqrt{2}} \\ & =\lim _{x \rightarrow \infty} \frac{x^3\left(\sqrt{1+x^4}-x^2\right)}{\sqrt{x^2+\sqrt{1+x^4}}+x \sqrt{2}} \\ & =\lim _{x \rightarrow \infty} \frac{x^3\left(1+x^4-x^4\right)}{\left(\sqrt{x^2+\sqrt{1+x^4}}+x \sqrt{2}\right)\left(\sqrt{1+x^4}+x^2\right)}\end{aligned}\)
\(\begin{aligned} & =\lim _{x \rightarrow \infty} \frac{x^3}{x^3\left(\sqrt{1+\sqrt{\frac{1}{x^4}+1}+\sqrt{2}}\right)\left(\sqrt{\frac{1}{x^4}+1}+1\right)} \\ & =\lim _{x \rightarrow \infty} \frac{1}{\left(\sqrt{1+\sqrt{\frac{1}{x^4}+1}+\sqrt{2}}\right)\left(\sqrt{\frac{1}{x^4}+1}+1\right)} \\ & =\frac{1}{(\sqrt{1+1}+\sqrt{2})(1+1)} \\ & =\frac{1}{4 \sqrt{2}}\end{aligned}\)