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  1. Prove that \(\displaystyle \lim_{x \to \infty} x \ln\!\left(1+\tfrac{1}{x}\right) = 1.\)
  2. By writing \(\displaystyle \left(1+\tfrac{1}{x}\right)^x = e^{\,x \ln(1+\tfrac{1}{x})}\) and using the fact that \(f(x)=e^x\) is continuous on \(\mathbb{R}\), prove that $$ \lim_{x \to \infty} \left(1+\tfrac{1}{x}\right)^x = e. $$

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