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Consider the function \(f(x)=\dfrac{\ln(1+x)}{x}\) for \(x>0\).
  1. Show that \(\displaystyle\lim_{x \to 0^+}\dfrac{\ln(1+x)}{x}\) exists and find its value.
  2. Hence, determine \(\displaystyle\lim_{x \to 0^+}\dfrac{\ln(1+2x)}{x}\).

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