\(\lim _{y \rightarrow 0} \frac{\sqrt{1+\sqrt{1+y^4}}-\sqrt{2}}{y^4}=\)

By rationalising, we get
\(\begin{aligned}
& \lim _{y \rightarrow 0} \frac{\sqrt{1+\sqrt{1+y^4}-\sqrt{2}}}{y^4} \\
& =\lim _{y \rightarrow 0} \frac{1+\sqrt{1+y^4}-2}{y^4\left(\sqrt{1+\sqrt{1+y^4}}+\sqrt{2}\right)} \\
& =\lim _{y \rightarrow 0} \frac{\sqrt{1+y^4}-1}{y^4\left(\sqrt{1+\sqrt{1+y^4}}+\sqrt{2}\right)} \\
& =\lim _{y \rightarrow 0} \frac{\sqrt{1+y^4}-1}{y^4\left(\sqrt{1+\sqrt{1+y^4}}+\sqrt{2}\right)} \times \frac{\sqrt{1+y^4}+1}{\sqrt{1+y^4}+1} \\
& =\lim _{y \rightarrow 0} \frac{y^4}{y^4\left(\sqrt{1+\sqrt{1+y^4}}+\sqrt{2}\right)\left(\sqrt{1+y^4}+1\right)} \\
& =\frac{1}{(\sqrt{1+\sqrt{1+0}+\sqrt{2})(\sqrt{1+0}+1)}}=\frac{1}{4 \sqrt{2}}
\end{aligned}\)