`lim_(x->oo)(sqrt(x^4+ax^3+3x^2+b x+2)-sqrt(x^4+2x^3+cx^2+3x-d))=4`

Answer

To solve the limit problem given by

\[ \lim_{x \to \infty} \left( \sqrt{x^4 + ax^3 + 3x^2 + bx + 2} - \sqrt{x^4 + 2x^3 + cx^2 + 3x - d} \right) = 4, \]

we will follow these steps:

### Step 1: Rationalization
We start by rationalizing the expression. We multiply and divide by the conjugate of the expression:

\[ \frac{\left( \sqrt{x^4 + ax^3 + 3x^2 + bx + 2} - \sqrt{x^4 + 2x^3 + cx^2 + 3x - d} \right) \left( \sqrt{x^4 + ax^3 + 3x^2 + bx + 2} + \sqrt{x^4 + 2x^3 + cx^2 + 3x - d} \right)}{\sqrt{x^4 + ax^3 + 3x^2 + bx + 2} + \sqrt{x^4 + 2x^3 + cx^2 + 3x - d}}. \]

### Step 2: Simplifying the Numerator
The numerator simplifies to:

\[ (x^4 + ax^3 + 3x^2 + bx + 2) - (x^4 + 2x^3 + cx^2 + 3x - d). \]

This simplifies to:

\[ (a - 2)x^3 + (3 - c)x^2 + (b - 3)x + (2 + d). \]

### Step 3: Simplifying the Denominator
The denominator becomes:

\[ \sqrt{x^4 + ax^3 + 3x^2 + bx + 2} + \sqrt{x^4 + 2x^3 + cx^2 + 3x - d}. \]

### Step 4: Dividing by \(x^2\)
Next, we divide the numerator and denominator by \(x^2\):

\[ \frac{(a - 2)x + (3 - c) + \frac{(b - 3)}{x} + \frac{(2 + d)}{x^2}}{\sqrt{1 + \frac{a}{x} + \frac{3}{x^2} + \frac{b}{x^3} + \frac{2}{x^4}} + \sqrt{1 + \frac{2}{x} + \frac{c}{x^2} + \frac{3}{x^3} - \frac{d}{x^4}}}. \]

### Step 5: Taking the Limit as \(x \to \infty\)
As \(x \to \infty\), the terms \(\frac{(b - 3)}{x}\) and \(\frac{(2 + d)}{x^2}\) vanish, and we are left with:

\[ \frac{(a - 2)x + (3 - c)}{2}. \]

### Step 6: Setting the Limit Equal to 4
For the limit to equal 4, we need:

\[ \frac{(a - 2)x + (3 - c)}{2} = 4. \]

This implies:

\[ (a - 2)x + (3 - c) = 8. \]

### Step 7: Finding Values of \(a\) and \(c\)
For this equation to hold as \(x \to \infty\), the coefficient of \(x\) must be zero:

1. \(a - 2 = 0 \implies a = 2\).
2. The constant term must equal 8: \(3 - c = 8 \implies c = -5\).

### Step 8: Conclusion
The values we have found are:

- \(a = 2\)
- \(c = -5\)

The values of \(b\) and \(d\) are not determined by the limit condition and can be any real numbers.