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GATE 2026 CH – Question 22

Engineering Mathematics · Calculus: Limits, continuity, differentiability, Taylor series and mean value theorem · 1 mark · Multiple choice

Which one of the following is the value of $\lim_{x \to 0} \frac{e^x - x - 1}{\cos x - 1}$ ?

  1. −1
  2. 0
  3. 1
  4. $\infty$

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Correct answer: (A) −1

Explanation

Expand for small $x$: $e^x - x - 1 \approx \frac{x^2}{2}$ and $\cos x - 1 \approx -\frac{x^2}{2}$. The ratio tends to $\frac{x^2/2}{-x^2/2} = -1$. (L'Hopital's rule applied twice gives the same value.)