GATE 2021 MN – Question 45
For a fixed real constant b, evaluate $\lim_{x\to\infty}\left(x\sqrt{x^2+b^2}-\sqrt{x^4+b^4}\right)$.
Practise this question in The GATE Grind →
Show answer and explanation
Correct answer: (B) $b^2/2$
Explanation
As x grows positive, $x\sqrt{x^2+b^2}=x^2+b^2/2+O(x^{-2})$. Also $\sqrt{x^4+b^4}=x^2+O(x^{-2})$. Subtracting cancels the leading terms and leaves b²/2, B. Both displayed radicals are square roots; the x⁴ term must not be read as a fourth root.