$\underset{y\to 0}{\lim }\frac{\sqrt{1+\sqrt{1+y^4}}-\sqrt{2}}{y^4}$

$$\begin{gathered} \underset{\mathrm{y}\to 0}{\mathrm{lt}}\frac{\sqrt{1+\sqrt{1+{\mathrm{y}}^4}-\sqrt{2}}}{{\mathrm{y}}^4}\times \frac{\sqrt{1+\sqrt{1+{\mathrm{y}}^4}+\sqrt{2}}}{\sqrt{1+\sqrt{1+{\mathrm{y}}^4}+\sqrt{2}}} \\ =\underset{\mathrm{y}\to 0}{\mathrm{lt}}\frac{1+\sqrt{1+{\mathrm{y}}^4}-2}{{\mathrm{y}}^4\left(\sqrt{1+\sqrt{1+{\mathrm{y}}^4}+\sqrt{2}}\right)} \\ =\underset{\mathrm{y}\to 0}{\mathrm{lt}}\frac{\sqrt{1+{\mathrm{y}}^4}-1}{{\mathrm{y}}^4\left(\sqrt{1+\sqrt{1+{\mathrm{y}}^4+\sqrt{2}}}\right)}\times \frac{\sqrt{1+{\mathrm{y}}^4}+1}{\sqrt{1+{\mathrm{y}}^4}+1} \\ =\underset{\mathrm{y}\to 0}{\mathrm{lt}}\frac{1+{\mathrm{y}}^4-1}{{\mathrm{y}}^4\left(\sqrt{1+\sqrt{1+{\mathrm{y}}^4}}+\sqrt{2}\right)\left(\sqrt{1+{\mathrm{y}}^4}+1\right)} \\ =\underset{\mathrm{y}\to 0}{\mathrm{lt}}\frac{\cancel{{\mathrm{y}}^4}}{\cancel{{\mathrm{y}}^4}\left(\sqrt{1+\sqrt{1+{\mathrm{y}}^4}}+\sqrt{2}\right)\left(\sqrt{1+{\mathrm{y}}^4}+1\right)} \\ =\frac{1}{\left(\sqrt{2}+\sqrt{2}\right)(1+1)}=\frac{1}{\left(2\sqrt{2}\right)2}=\frac{1}{4\sqrt{2}} \end{gathered}$$