Limits of sequences

Task number: 2870

Determine the following limits, or prove that they do not exist.

  • Variant 1

    \(\displaystyle \lim_{n\to\infty} \frac{2n^2+4n+n \sin n}{ n \cos 3n + (2n +\sin n)^2}\)

  • Variant 2

    \(\displaystyle \lim_{n\to\infty} \frac{\lfloor \sqrt{n} \rfloor}{\sqrt{n}}\)

  • Variant 3

    \(\displaystyle \lim_{n\to\infty} \cos (n^2 \pi) +\cos ((n+1) \pi) \)

  • Variant 4

    \(\displaystyle \lim_{n\to\infty} n^{\cos (n \pi)} \)

  • Variant 5

    \(\displaystyle \lim_{n\to\infty} \frac{\sqrt[4]{n^5+2}-\sqrt[3]{n^2+1}}{\sqrt[5]{n^4+2}-\sqrt{n^3+1}}\)

  • Variant 6

    \(\displaystyle \lim_{n\to\infty} \frac{3^n+n^5}{n^6+n!}\)

  • Variant 7

    \(\displaystyle \lim_{n\to\infty} \frac{\sqrt[3]{n^2}\sin(n!)}{n+1}\)

Difficulty level: Moderate task
Routine calculation training
Cs translation
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