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GATE 2016 ME (ME3) – Question 12

Engineering Mathematics · Calculus: Limit, continuity, differentiability, mean value theorems and indeterminate forms · 1 mark · Multiple choice

The value of $\lim_{x\to0}\ln(1+4x)/(e^{3x}-1)$ is ______.

  1. 0
  2. 1/12
  3. 4/3
  4. 1

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Correct answer: (C) 4/3

Explanation

Both numerator and denominator vanish at zero. Using ln(1+4x)=4x+O(x²) and exp(3x)−1=3x+O(x²), the ratio tends to **4/3, C**. Equivalently, one application of l’Hôpital’s rule gives the same result.