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GATE 2023 CH – Question 11

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

Which one of the following is the CORRECT value of $y$, as defined by the expression given below? $y = \lim_{x \to 0}\frac{2x}{e^x - 1}$

  1. 1
  2. 2
  3. 0
  4. ∞

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Correct answer: (B) 2

Explanation

Near $x = 0$, $e^x - 1 \approx x$, so $\frac{2x}{e^x - 1} \to 2$ (by L'Hopital's rule, $\frac{2}{e^x} \to 2$).