Limit Evaluation: Step-by-Step Explanation

We are evaluating the limit: $\underset{y\to 0}{\lim }\frac{\sqrt{1+\sqrt{1+y^4}}-\sqrt{2}}{y^4}$

This is a $\frac{0}{0}$ indeterminate form. We can solve it using rationalization and standard limit formulas.

Step 1: Rationalize the Numerator

Multiply the numerator and denominator by the conjugate of the numerator to simplify the expression.

Conjugate: $\sqrt{1+\sqrt{1+y^4}}+\sqrt{2}$

Limit becomes: $\underset{y\to 0}{\lim }\frac{(\sqrt{1+\sqrt{1+y^4}}-\sqrt{2})(\sqrt{1+\sqrt{1+y^4}}+\sqrt{2})}{y^4(\sqrt{1+\sqrt{1+y^4}}+\sqrt{2})}$

Step 2: Simplify the Numerator

Using the identity $(a-b)(a+b)=a^2-b^2$, the numerator simplifies to:

$(1+\sqrt{1+y^4})-\left(2\right)=\sqrt{1+y^4}-1$

So the limit is now: $\underset{y\to 0}{\lim }\frac{\sqrt{1+y^4}-1}{y^4(\sqrt{1+\sqrt{1+y^4}}+\sqrt{2})}$

Step 3: Rationalize Again

The new numerator $\sqrt{1+y^4}-1$ is also a $\frac{0}{0}$ form. Multiply numerator and denominator by its conjugate $\sqrt{1+y^4}+1$.

Limit becomes: $\underset{y\to 0}{\lim }\frac{(\sqrt{1+y^4}-1)(\sqrt{1+y^4}+1)}{y^4(\sqrt{1+\sqrt{1+y^4}}+\sqrt{2})(\sqrt{1+y^4}+1)}$

Step 4: Simplify the New Numerator

Using the identity again: $(\sqrt{1+y^4}-1)(\sqrt{1+y^4}+1)=(1+y^4)-1=y^4$

The limit simplifies to: $\underset{y\to 0}{\lim }\frac{y^4}{y^4(\sqrt{1+\sqrt{1+y^4}}+\sqrt{2})(\sqrt{1+y^4}+1)}=\underset{y\to 0}{\lim }\frac{1}{(\sqrt{1+\sqrt{1+y^4}}+\sqrt{2})(\sqrt{1+y^4}+1)}$

Step 5: Evaluate the Limit by Direct Substitution

Now we can substitute $y=0$ directly into the simplified expression.