Find limit $\lim\limits _{x\rightarrow \infty }\left(\sqrt{x^{4} +x^{2}\sqrt{x^{4} +1}} -\sqrt{2x^{4}}\right)$ $\displaystyle \lim\limits _{x\rightarrow \infty }\left(\sqrt{x^{4} +x^{2}\sqrt{x^{4} +1}}\displaystyle -\sqrt{2x^{4}}\right)$
$\displaystyle \lim\limits _{x\rightarrow \infty }\left(\sqrt{x^{4} +x^{2}\sqrt{x^{4} +1}} -\sqrt{2x^{4}}\right)$$\displaystyle =\displaystyle \lim\limits _{x\rightarrow \infty }\dfrac{\left(\sqrt{x^{4} +x^{2}\sqrt{x^{4} +1}} -\sqrt{2x^{4}}\right)\left(\sqrt{x^{4} +x^{2}\sqrt{x^{4} +1}} +\sqrt{2x^{4} \ }\right)}{\sqrt{x^{4} +x^{2}\sqrt{x^{4} +1}} +\sqrt{2x^{4} \ }} =$
$\displaystyle =\lim\limits _{x\rightarrow \infty }\dfrac{x^{2}\sqrt{x^{4} +1} -x^{4}}{\sqrt{x^{4} +x^{2}\sqrt{x^{4} +1}} +\sqrt{2x^{4} \ }}$
What is the next step should be? Please help! 
 A: Now, use that
$$\sqrt{x^4+1}-x^2=\frac{1}{\sqrt{x^4+1}+x^2}.$$
A: Use the trick twice for the numerator to obtain
$$\dfrac{x^{2}\sqrt{x^{4} +1} -x^{4}}{\sqrt{x^{4} +x^{2}\sqrt{x^{4} +1}} +\sqrt{2x^{4} \ }}=\dfrac{x^{2}\sqrt{x^{4} +1} -x^{4}}{\sqrt{x^{4} +x^{2}\sqrt{x^{4} +1}} +\sqrt{2x^{4} \ }}\cdot \dfrac{x^{2}\sqrt{x^{4} +1} +x^{4}}{x^{2}\sqrt{x^{4} +1} +x^{4}}=$$
$$=\dfrac{x^4}{\left(\sqrt{x^{4} +x^{2}\sqrt{x^{4} +1}} +\sqrt{2x^{4} \ }\right)\left(x^{2}\sqrt{x^{4} +1} +x^{4}\right)}\sim\frac{1}{4\sqrt 2 x^2}\to 0$$
A: $$
\sqrt{x^{4} +x^{2}\sqrt{x^{4} +1}}-\sqrt{2x^4}=
\dfrac{x^{2}\sqrt{x^{4} +1} -x^{4}}{\sqrt{x^{4} +x^{2}\sqrt{x^{4} +1}} +\sqrt{2x^{4}}}=
\dfrac{x^{2}\sqrt{x^{4} +1} -x^{4}}{\sqrt{x^{4} +x^{2}\sqrt{x^{4} +1}} +\sqrt{2x^{4} \ }}\cdot\frac{x^{2}\sqrt{x^{4} +1} +x^{4}}{x^{2}\sqrt{x^{4} +1} +x^{4}}\\=\frac{x^4}{\big(\sqrt{x^{4} +x^{2}\sqrt{x^{4} +1}} +\sqrt{2x^{4}}\big)\big(x^{2}\sqrt{x^{4} +1}+x^4\big)}\\=\frac{1}{\big(\sqrt{1 +\sqrt{1+x^{-4}}} +\sqrt{2}\big)\big(\sqrt{x^{4}+1}+x^2\big)}\,\to\,0
$$
A: Another approach is using substitution $x=\cot t$:
\begin{align}
\lim_{x\to\infty}\left(\sqrt{x^{4} +x^{2}\sqrt{x^{4} +1}}-\sqrt{2x^{4}}\right) 
&=\lim_{t\to0}\dfrac{\sqrt{\cos^2t+\cos t}-\sqrt{2\cos^2t}}{\sin t} \\
&=\lim_{t\to0}\dfrac{\cos t(1-\cos t)}{\sin t(\sqrt{\cos^2t+\cos t}+\sqrt{2\cos^2t})} \\
&=\lim_{t\to0}\dfrac{\cos t}{\sqrt{\cos^2t+\cos t}+\sqrt{2\cos^2t}}\dfrac{2\sin^2\frac{t}{2}}{\frac{t^2}{2}}\dfrac{t}{\sin t}\dfrac{t}{2} \\
&=\color{blue}{0}
\end{align}
