Given that `lim_(xtooo)((2+x^(2))/(1+x)-Ax-B)=3`
What is the value of B ?

Options

A. `-1`
B. 3
C. `-4`
D. `-3`

Answer

To solve the limit problem given by

\[ \lim_{x \to \infty} \left( \frac{2 + x^2}{1 + x} - Ax - B \right) = 3, \]

we need to find the value of \( B \).

### Step-by-step Solution:

1. **Set up the limit expression**:
We start with the expression inside the limit:
\[ \frac{2 + x^2}{1 + x} - Ax - B. \]

2. **Combine the terms**:
To combine the terms, we can express everything over a common denominator:
\[ \frac{2 + x^2 - (Ax + B)(1 + x)}{1 + x}. \]

3. **Expand the numerator**:
Expanding the numerator gives:
\[ 2 + x^2 - (Ax + B + Ax^2 + Bx) = (1 - A)x^2 + (2 - B - A)x + 2 - B. \]

4. **Analyze the degrees of the polynomial**:
The highest degree term in the numerator is \((1 - A)x^2\). For the limit to exist and be finite as \( x \to \infty \), the coefficient of \( x^2 \) must be zero:
\[ 1 - A = 0 \implies A = 1. \]

5. **Substitute \( A \) back into the limit**:
Now substituting \( A = 1 \) back into the expression, we have:
\[ \lim_{x \to \infty} \left( \frac{(2 - B)x + 2 - B}{1 + x} \right). \]

6. **Simplify the limit**:
As \( x \to \infty \), the dominant term in the numerator is \((2 - B)x\) and in the denominator is \(x\):
\[ \lim_{x \to \infty} \frac{(2 - B)x}{x} = 2 - B. \]

7. **Set the limit equal to 3**:
We know from the problem statement that this limit equals 3:
\[ 2 - B = 3. \]

8. **Solve for \( B \)**:
Rearranging gives:
\[ -B = 3 - 2 \implies -B = 1 \implies B = -1. \]

### Final Answer:
Thus, the value of \( B \) is:
\[ \boxed{-1}. \]