$\underset{x\to \infty }{\lim }{x}^3\left[\sqrt{x^2+\sqrt{x^4+1}}-\sqrt{2}x\right]=$

$L=\underset{x\to \infty }{\lim }{x}^3\left[\sqrt{x^2+\sqrt{x^4+1}}-\sqrt{2}x\right]$

On rationalization

$L=\underset{x\to \infty }{\lim }{x}^3\left[\sqrt{x^2+\sqrt{x^4+1}}-\sqrt{2}x\right]\times \frac{\left[\sqrt{x^2+\sqrt{x^4+1}}+\sqrt{2}x\right]}{\left[\sqrt{x^2+\sqrt{x^4+1}}+\sqrt{2}x\right]}$

$=\underset{x\to \infty }{\lim }{x}^3\frac{\left[x^2+\sqrt{x^4+1}-2x^2\right]}{\left[\sqrt{x^2+\sqrt{x^4+1}}+\sqrt{2}x\right]}$

$=\underset{x\to \infty }{\lim }{x}^3\frac{\left[\sqrt{x^4+1}-x^2\right]}{\left[\sqrt{x^2+\sqrt{x^4+1}}+\sqrt{2}x\right]}\times \frac{\sqrt{x^4+1}+x^2}{\sqrt{x^4+1}+x^2}$

$=\underset{x\to \infty }{\lim }{x}^3\frac{1}{\left[\sqrt{x^2+\sqrt{x^4+1}}+\sqrt{2}x\right]}\times \frac{x^4+1-x^4}{\sqrt{x^4+1}+x^2}$

$=\underset{x\to \infty }{\lim }{x}^3\frac{1}{\left[\sqrt{x^2+\sqrt{x^4+1}}+\sqrt{2}x\right]}\times \frac{1}{\sqrt{x^4+1}+x^2}$

$=\underset{x\to \infty }{\lim }\frac{1}{\left[\sqrt{1+\sqrt{1+{\displaystyle \frac{1}{x^4}}}}+\sqrt{2}\right]}\times \frac{1}{\sqrt{1+{\displaystyle \frac{1}{x^4}}}+1}$

$=\frac{1}{\left[\sqrt{1+\sqrt{1+0}}+\sqrt{2}\right]}\times \frac{1}{\sqrt{1+{\displaystyle 0}}+1}=\frac{1}{2\sqrt{2}}\times \frac{1}{2}$

$=\frac{1}{4\sqrt{2}}$