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A limit that hides a difference of logs

EasyCalculus~3m

Evaluate lim⁡x→∞x(ln⁡(x+1)−ln⁡x)\displaystyle\lim_{x\to\infty} x\big(\ln(x+1)-\ln x\big). (to two decimals)

Show hints (2)+
  1. Combine the two logs into ln⁡(1+1/x)\ln(1+1/x) first - then it's ∞⋅0\infty\cdot 0.
  2. Substitute t=1/xt=1/x to get ln⁡(1+t)/t→1\ln(1+t)/t\to 1 as t→0t\to 0.

Answer

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1 (± 0.01)

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Asked at: Timed Mental-Math & Sequences, Game-Based Aptitude

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