Maths · Limits, continuity and differentiability

Arrange the following limits in the ascending order: (1) ^{x+2} \) (2) ^{3 / x}

Arrange the following limits in the ascending order: (1) \( \lim _{x \rightarrow \infty}\left(\frac{1+x}{2+x}\right)^{x+2} \) (2) \( \lim _{x \rightarrow 0}(1+2 x)^{3 / x} \) (3) \( \lim _{\boldsymbol{\theta} \rightarrow \mathbf{0}} \frac{\sin \boldsymbol{\theta}}{\mathbf{2} \boldsymbol{\theta}} \) (4) \( \lim _{x \rightarrow 0} \frac{\log _{e}(1+x)}{x} \)

  • A. 1,2,3,4
  • B. 1,3,4,2
  • C. 1,4,3,2
  • D. 3,4,1,2

Step-by-step solution

Compute each limit: (1) = 1/e ≈ 0.3679, (2) = e^6 ≈ 403.43, (3) = 1/2 = 0.5, (4) = 1. Ascending order: (1) < (3) < (4) < (2) → 1,3,4,2.
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