If A,B,C are positive real numbers such that `lim_(xrarr oo) (sqrt(Ax^2+Bx)-Cx)=2, then (BC)/(A)` equals

Options

A. 4
B. 2
C. `(1)/(2)`
D. `none of these

Explanation

To solve the problem, we need to find the value of \(\frac{BC}{A}\) given that

\[ \lim_{x \to \infty} \left( \sqrt{Ax^2 + Bx} - Cx \right) = 2 \]

### Step-by-Step Solution:

1. **Rewrite the Limit Expression**:
We start with the expression inside the limit:
\[ \sqrt{Ax^2 + Bx} - Cx \]
To simplify this, we can rationalize it by multiplying and dividing by the conjugate:
\[ \frac{(\sqrt{Ax^2 + Bx} - Cx)(\sqrt{Ax^2 + Bx} + Cx)}{\sqrt{Ax^2 + Bx} + Cx} \]
This gives us:
\[ \frac{(Ax^2 + Bx) - C^2x^2}{\sqrt{Ax^2 + Bx} + Cx} \]

2. **Simplify the Numerator**:
The numerator simplifies to:
\[ (A - C^2)x^2 + Bx \]
Thus, we have:
\[ \frac{(A - C^2)x^2 + Bx}{\sqrt{Ax^2 + Bx} + Cx} \]

3. **Analyze the Limit**:
As \(x\) approaches infinity, the dominant term in the numerator is \((A - C^2)x^2\) and in the denominator, it is \(\sqrt{Ax^2}\) which behaves like \(\sqrt{A}x\). Therefore, we can write:
\[ \lim_{x \to \infty} \frac{(A - C^2)x^2 + Bx}{\sqrt{Ax^2 + Bx} + Cx} = \lim_{x \to \infty} \frac{(A - C^2)x^2}{\sqrt{A}x + Cx} \]
Simplifying further, we get:
\[ \lim_{x \to \infty} \frac{(A - C^2)x}{\sqrt{A} + C} \]

4. **Set the Limit Equal to 2**:
For the limit to equal 2, we need:
\[ \frac{(A - C^2)}{\sqrt{A} + C} = 2 \]

5. **Solve for \(A\) and \(C\)**:
Rearranging gives us:
\[ A - C^2 = 2(\sqrt{A} + C) \]
This can be rearranged to:
\[ A - 2\sqrt{A} - 2C - C^2 = 0 \]

6. **Use the Quadratic Formula**:
Treating this as a quadratic in terms of \(\sqrt{A}\):
\[ (\sqrt{A})^2 - 2\sqrt{A} - (2C + C^2) = 0 \]
Using the quadratic formula, we find:
\[ \sqrt{A} = \frac{2 \pm \sqrt{4 + 4(2C + C^2)}}{2} = 1 \pm \sqrt{2 + C^2} \]
Since \(A\) is positive, we take the positive root:
\[ \sqrt{A} = 1 + \sqrt{2 + C^2} \]

7. **Find the Ratio \(\frac{BC}{A}\)**:
From the limit condition, we also have:
\[ \frac{B}{C} = 4 \implies B = 4C \]
Now substituting \(A = C^2\):
\[ \frac{BC}{A} = \frac{(4C)C}{C^2} = \frac{4C^2}{C^2} = 4 \]

### Final Answer:
Thus, the value of \(\frac{BC}{A}\) is:
\[ \boxed{4} \]