The GATE Grind

GATE 2025 CE (CE1) – Question 36

Engineering Mathematics · Calculus: Limits, continuity, mean value theorems, maxima and minima, Taylor series · 2 marks · Multiple choice

The value of $\lim_{x \to \infty}\left(x - \sqrt{x^2 + x}\right)$ is equal to

  1. -1
  2. -0.5
  3. -2
  4. 0

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (B) -0.5

Explanation

Multiply and divide by the conjugate: $x - \sqrt{x^2 + x} = \frac{x^2 - (x^2 + x)}{x + \sqrt{x^2 + x}} = \frac{-x}{x + \sqrt{x^2 + x}} = \frac{-1}{1 + \sqrt{1 + 1/x}}$. As $x \to \infty$ this tends to $\frac{-1}{1 + 1} = -0.5$.