Animated Solution for Mathematics - Limits, Continuity and Differentiability: ${\lim }_{y\to 0}\frac{\sqrt{1+\sqrt{1+y^4}}-\sqrt{2}}{y^4}$

We are evaluating the limit:

Upon direct substitution of $y=0$, the expression yields $\frac{\sqrt{1+1}-\sqrt{2}}{0}=\frac{0}{0}$. This confirms an indeterminate form, signaling that a factor of $y^4$ must be extracted from the numerator to cancel the denominator.

To eliminate the outer radical, we multiply the numerator and the denominator by the conjugate $\sqrt{1+\sqrt{1+y^4}}+\sqrt{2}$. Applying the identity $(a-b)(a+b)=a^2-b^2$, the numerator becomes:

The limit expression now stands as:

Substituting $y=0$ again results in another $0/0$ form. We must rationalize the remaining radical $\sqrt{1+y^4}-1$ by multiplying the numerator and denominator by $\sqrt{1+y^4}+1$. The numerator simplifies as follows:

This step successfully isolates the $y^4$ term. The expression is now:

We cancel the $y^4$ terms from the numerator and the denominator. We are left with:

Now, we perform direct substitution by setting $y=0$: 1. The first bracket: $\sqrt{1+\sqrt{1+0}}+\sqrt{2}=\sqrt{2}+\sqrt{2}=2\sqrt{2}$ 2. The second bracket: $\sqrt{1+0}+1=1+1=2$

Multiplying these results, we obtain $2\sqrt{2}\cdot 2=4\sqrt{2}$. Therefore, the final value of the limit is: