Chapter 5.4 Practice

Questions

  1. Suppose that \((\theta_n)\) is a null sequence with only nonzero terms. Calculate \[\lim_{n\to\infty} \tfrac{\sin(5\theta_n)}{6\theta_n}.\]

  2. Suppose that \((\theta_n)\) is a null sequence with only nonzero terms. Calculate \[\lim_{n\to\infty} \tfrac{3-3\cos(\theta_n)}{\theta_n}.\]

  3. Calculate the following limits. Carefully justify your reasoning.

    1. \(\displaystyle\lim_{n\to\infty} \frac{\sin(\frac{3}{n})}{\frac{6}{n}}\)
    2. \((\theta_n)\) is a null sequence with only nonzero terms; \(\displaystyle\lim_{n\to\infty} \frac{\tan(\theta_n)}{\theta_n}.\)
    3. \(\displaystyle\lim_{n\to\infty} \frac{\sin(-\frac{3}{n^2})}{\frac{8}{n^2}}\)
    4. \(\displaystyle\lim_{n\to\infty} \frac{\sin(-7e^{-n})}{\sin(-6e^{-n})}\)
    5. \(\displaystyle\lim_{n\to\infty} \frac{1-\cos\left(\frac{7}{n}\right)}{\frac{1}{n^2}}\)

Answers

  1. \(\frac{5}{6}\)

  2. \(0\)

    1. \(\frac{1}{2}\)
    2. \(1\)
    3. \(-\frac{3}{8}\)
    4. \(-\frac{7}{6}\)
    5. \(\frac{49}{2}\)

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