Question

The value of \[ \lim_{x \to \infty} \left(x \sqrt{x^2 + b^2} - \sqrt{x^4 + b^4}\right) \text{ is} \]

• A. 0.
• B. \( \dfrac{b^2}{2} \)
• C. \( \infty \)
• D. \( b^2 \)

Hint:

To evaluate limits of expressions involving square roots, simplify the terms using binomial expansions or approximations for large values of \( x \).

Answer 1

The Correct Option is B

We are asked to evaluate the limit of the given expression as \( x \to \infty \). Let's simplify the expression step by step: We start with the given expression: \[ \lim_{x \to \infty} \left(x \sqrt{x^2 + b^2} - \sqrt{x^4 + b^4}\right). \] Step 1: Simplifying the terms The term \( x \sqrt{x^2 + b^2} \) can be approximated for large \( x \) by factoring out \( x^2 \) from the square root: \[ x \sqrt{x^2 + b^2} = x \sqrt{x^2(1 + \frac{b^2}{x^2})} = x^2 \sqrt{1 + \frac{b^2}{x^2}}. \] For large \( x \), we can expand \( \sqrt{1 + \frac{b^2}{x^2}} \) as: \[ \sqrt{1 + \frac{b^2}{x^2}} \approx 1 + \frac{b^2}{2x^2}. \] So, \[ x \sqrt{x^2 + b^2} \approx x^2 + \frac{b^2}{2x}. \] Step 2: Simplifying \( \sqrt{x^4 + b^4} \) Now, consider the second term \( \sqrt{x^4 + b^4} \). For large \( x \), we can approximate it as: \[ \sqrt{x^4 + b^4} = x^2 \sqrt{1 + \frac{b^4}{x^4}} \approx x^2 + \frac{b^4}{2x^2}. \] Step 3: Subtract the two terms Now subtract the two expressions: \[ x \sqrt{x^2 + b^2} - \sqrt{x^4 + b^4} \approx \left(x^2 + \frac{b^2}{2x}\right) - \left(x^2 + \frac{b^4}{2x^2}\right). \] This simplifies to: \[ \frac{b^2}{2x} - \frac{b^4}{2x^2}. \] As \( x \to \infty \), the second term \( \frac{b^4}{2x^2} \) tends to zero. Thus, the dominant term is: \[ \frac{b^2}{2x}. \] Therefore, the limit is \( \frac{b^2}{2} \). Thus, the correct answer is (B).