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NDA 2020 Maths · Mathematics · Limits · medium

As x tends to infinity, the expression √(x^2+5x)-x approaches the value

Single correct option

As xx tends to infinity, the expression x2+5x−x\sqrt{x^2+5x}-x approaches the value
  1. A. 00
  2. B. 55
  3. C. 12\dfrac12
  4. D. 52\dfrac52
Show answer and solution

Answer: (D) 52\dfrac52

This is an ∞−∞\infty-\infty form, so rationalise: x2+5x−x=5xx2+5x+x=51+5/x+1→52\sqrt{x^2+5x}-x=\dfrac{5x}{\sqrt{x^2+5x}+x}=\dfrac{5}{\sqrt{1+5/x}+1}\to\dfrac52. The answer 00 comes from wrongly cancelling x2\sqrt{x^2} against xx and discarding the 5x5x term, which is of the same order as xx. The value 12\dfrac12 is the limit of x(x2+1−x)x\left(\sqrt{x^2+1}-x\right), a different expression.
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Original question written in the NDA 2020 Maths pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake