Calculus/Limits/Exercises

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  • \lim_{x \to \infty} \left [ \cos{\left ( \frac 1 x \right )} \right ]^{x^3 \log{(1 + {{1}\over{x}})}}
  • \lim_{x \to \infty} \frac{1}{x} in \mathbb{R}^+
Answer: does not exist.
  • \lim_{n \to \infty} \frac{{\left (n+1 \right )}^\alpha - n^\alpha}{n^{\alpha-1}}
Answer: α
  • \lim_{x \to \infty} \sqrt{x} \log {\left ( 1 + e^x \right )} - x \sqrt{x}
Answer: 0
  • \lim_{x \to \infty} \sqrt{x} \log {\left ( 1 + e^x \right )} - x \sqrt{x-1}
Answer: -\infty
  • \lim_{x \to \infty} \frac{\sqrt[x]{x+\sin{x}} \left (2+\sin{x} \right )^x}{x!}
  • \lim_{x \to \infty} \frac{\sqrt{x^3+x}-x \sqrt{x}}{x+6 \sin{x}}
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