Limits
Rationalization of nested surd; limit as $y\to\infty$
MJMT_Full_Test_09
Grade 12
Question:
$\displaystyle\lim_{y\to\infty}\frac{\sqrt{1+\sqrt{1+y^4}}-\sqrt2}{y^4}$
$4\sqrt2$
$\dfrac{1}{4\sqrt2}$
$\sqrt2$
0
Step-by-Step Solution
Key Concept: Rationalize repeatedly. $\sqrt{1+y^4}\approx y^2$ for large $y$. Then $\sqrt{1+y^2}\approx y$. Each rationalization step removes one layer, giving $1/(4\sqrt{2})$.
Limit $=\dfrac{1}{4\sqrt2}$.
Correct Answer: 2