limy→0 1+1+y4−2y4

limy→0 1+1+y4−2y4=limy→0 1+1+y4−2y41+1+y4+2=limy→0 1+y4−11+y4+1y41+1+y4+21+y4+1=limy→0 1+y4−1y41+1+y4+21+y4+1=limy→0 11+1+y4+21+y4+1=142

limy→0 1+1+y4−2y4

limy→0 1+1+y4−2y4=limy→0 1+1+y4−2y41+1+y4+2=limy→0 1+y4−11+y4+1y41+1+y4+21+y4+1=limy→0 1+y4−1y41+1+y4+21+y4+1=limy→0 11+1+y4+21+y4+1=142

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

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exists and equals $\frac{1}{4 \sqrt{2}}$

does not exist

exists and equals $\frac{1}{2 \sqrt{2}}$

exists and equals $\frac{1}{2 \sqrt{2 (\sqrt{2} + 1)}}$

answer is A.

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Detailed Solution

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

$\begin{array}{l} = lim_{y \to 0} \frac{1 + \sqrt{1 + y^{4}} - 2}{y^{4} (\sqrt{1 + \sqrt{1 + y^{4}}} + \sqrt{2})} \\ = lim_{y \to 0} \frac{(\sqrt{1 + y^{4}} - 1) (\sqrt{1 + y^{4}} + 1)}{y^{4} (\sqrt{1 + \sqrt{1 + y^{4}}} + \sqrt{2}) (\sqrt{1 + y^{4}} + 1)} \end{array}$

$\begin{array}{l} = lim_{y \to 0} \frac{1 + y^{4} - 1}{y^{4} (\sqrt{1 + \sqrt{1 + y^{4}}} + \sqrt{2}) (\sqrt{1 + y^{4}} + 1)} \\ = lim_{y \to 0} \frac{1}{(\sqrt{1 + \sqrt{1 + y^{4}}} + \sqrt{2}) (\sqrt{1 + y^{4}} + 1)} = \frac{1}{4 \sqrt{2}} \end{array}$