Evaluate `lim_(xtooo) x^(3){sqrt(x^(2)+sqrt(1+x^(4)))-xsqrt(2)}.`

Explanation

To evaluate the limit

\[ \lim_{x \to \infty} x^3 \left( \sqrt{x^2 + \sqrt{1 + x^4}} - x \sqrt{2} \right), \]

we will follow these steps:

### Step 1: Rewrite the expression
We start with the limit expression:

\[ \lim_{x \to \infty} x^3 \left( \sqrt{x^2 + \sqrt{1 + x^4}} - x \sqrt{2} \right). \]

### Step 2: Multiply by the conjugate
To simplify the expression, we multiply and divide by the conjugate:

\[ \lim_{x \to \infty} \frac{x^3 \left( \sqrt{x^2 + \sqrt{1 + x^4}} - x \sqrt{2} \right) \left( \sqrt{x^2 + \sqrt{1 + x^4}} + x \sqrt{2} \right)}{\sqrt{x^2 + \sqrt{1 + x^4}} + x \sqrt{2}}. \]

This gives us:

\[ \lim_{x \to \infty} \frac{x^3 \left( \left( \sqrt{x^2 + \sqrt{1 + x^4}} \right)^2 - \left( x \sqrt{2} \right)^2 \right)}{\sqrt{x^2 + \sqrt{1 + x^4}} + x \sqrt{2}}. \]

### Step 3: Simplify the numerator
The numerator simplifies to:

\[ \left( x^2 + \sqrt{1 + x^4} - 2x^2 \right) = \sqrt{1 + x^4} - x^2. \]

Thus, we have:

\[ \lim_{x \to \infty} \frac{x^3 \left( \sqrt{1 + x^4} - x^2 \right)}{\sqrt{x^2 + \sqrt{1 + x^4}} + x \sqrt{2}}. \]

### Step 4: Simplify \(\sqrt{1 + x^4}\)
For large \(x\), we can approximate \(\sqrt{1 + x^4}\) as:

\[ \sqrt{1 + x^4} \approx x^2 \sqrt{1 + \frac{1}{x^4}} \approx x^2 \left( 1 + \frac{1}{2x^4} \right) = x^2 + \frac{1}{2x^2}. \]

So,

\[ \sqrt{1 + x^4} - x^2 \approx \frac{1}{2x^2}. \]

### Step 5: Substitute back into the limit
Now substituting back, we get:

\[ \lim_{x \to \infty} \frac{x^3 \cdot \frac{1}{2x^2}}{\sqrt{x^2 + \sqrt{1 + x^4}} + x \sqrt{2}} = \lim_{x \to \infty} \frac{\frac{x}{2}}{\sqrt{x^2 + \sqrt{1 + x^4}} + x \sqrt{2}}. \]

### Step 6: Simplify the denominator
For large \(x\), \(\sqrt{x^2 + \sqrt{1 + x^4}} \approx x\), thus:

\[ \sqrt{x^2 + \sqrt{1 + x^4}} + x \sqrt{2} \approx x + x \sqrt{2} = x(1 + \sqrt{2}). \]

### Step 7: Final limit calculation
Now we have:

\[ \lim_{x \to \infty} \frac{\frac{x}{2}}{x(1 + \sqrt{2})} = \lim_{x \to \infty} \frac{1}{2(1 + \sqrt{2})} = \frac{1}{2(1 + \sqrt{2})}. \]

### Final Answer
Thus, the limit evaluates to:

\[ \frac{1}{2(1 + \sqrt{2})}. \]