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The squeeze theorem practice problems

Trap an oscillating expression between two bounds with the same limit. 60 problems with worked solutions; in 60 of them every equation is proved by a computer algebra system.

16 worked problems

Try each one before opening the solution. Every line is labeled with what checked it. How we verify

Problem 1.621

medium
Use the squeeze theorem to find \( \displaystyle \lim_{x \to \infty} -1 + \frac{\cos{\left(2 x \right)}}{x} \).
  1. Since −1 ≤ cos(·) ≤ 1, the expression lies between -1 - 1/x and -1 + 1/x.
    Reviewed
  2. \[ \lim_{x \to \infty}\left(-1 - \frac{1}{x}\right) = -1 \]
    The lower bound's limit.✓ Proved
  3. \[ \lim_{x \to \infty}\left(-1 + \frac{1}{x}\right) = -1 \]
    The upper bound's limit, the same.✓ Proved
  4. Both bounds tend to the same value, so the expression trapped between them does too.
    Reviewed
Answer \( -1 \)

✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 1.623

medium
Use the squeeze theorem to find \( \displaystyle \lim_{x \to 0} x \sin{\left(\frac{4}{x} \right)} \).
  1. Since −1 ≤ sin(·) ≤ 1, the expression lies between -Abs(x) and Abs(x).
    Reviewed
  2. \[ \lim_{x \to 0}\left(- \left|{x}\right|\right) = 0 \]
    The lower bound's limit.✓ Proved
  3. \[ \lim_{x \to 0} \left|{x}\right| = 0 \]
    The upper bound's limit, the same.✓ Proved
  4. Both bounds tend to the same value, so the expression trapped between them does too.
    Reviewed
Answer \( 0 \)

✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 1.624

medium
Use the squeeze theorem to find \( \displaystyle \lim_{x \to 0} x^{2} \sin{\left(\frac{4}{x} \right)} \).
  1. Since −1 ≤ sin(·) ≤ 1, the expression lies between -x**2 and x**2.
    Reviewed
  2. \[ \lim_{x \to 0}\left(- x^{2}\right) = 0 \]
    The lower bound's limit.✓ Proved
  3. \[ \lim_{x \to 0} x^{2} = 0 \]
    The upper bound's limit, the same.✓ Proved
  4. Both bounds tend to the same value, so the expression trapped between them does too.
    Reviewed
Answer \( 0 \)

✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 1.626

medium
Use the squeeze theorem to find \( \displaystyle \lim_{n \to \infty} \frac{\cos{\left(2 n \right)}}{n} \).
  1. Since −1 ≤ cos(·) ≤ 1, the expression lies between -1/n and 1/n.
    Reviewed
  2. \[ \lim_{n \to \infty}\left(- \frac{1}{n}\right) = 0 \]
    The lower bound's limit.✓ Proved
  3. \[ \lim_{n \to \infty} \frac{1}{n} = 0 \]
    The upper bound's limit, the same.✓ Proved
  4. Both bounds tend to the same value, so the expression trapped between them does too.
    Reviewed
Answer \( 0 \)

✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 1.627

medium
Use the squeeze theorem to find \( \displaystyle \lim_{x \to 0} x^{2} \cos{\left(\frac{2}{x} \right)} \).
  1. Since −1 ≤ cos(·) ≤ 1, the expression lies between -x**2 and x**2.
    Reviewed
  2. \[ \lim_{x \to 0}\left(- x^{2}\right) = 0 \]
    The lower bound's limit.✓ Proved
  3. \[ \lim_{x \to 0} x^{2} = 0 \]
    The upper bound's limit, the same.✓ Proved
  4. Both bounds tend to the same value, so the expression trapped between them does too.
    Reviewed
Answer \( 0 \)

✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 1.628

medium
Use the squeeze theorem to find \( \displaystyle \lim_{x \to \infty} \frac{\cos{\left(4 x \right)}}{x} \).
  1. Since −1 ≤ cos(·) ≤ 1, the expression lies between -1/x and 1/x.
    Reviewed
  2. \[ \lim_{x \to \infty}\left(- \frac{1}{x}\right) = 0 \]
    The lower bound's limit.✓ Proved
  3. \[ \lim_{x \to \infty} \frac{1}{x} = 0 \]
    The upper bound's limit, the same.✓ Proved
  4. Both bounds tend to the same value, so the expression trapped between them does too.
    Reviewed
Answer \( 0 \)

✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 1.629

medium
Use the squeeze theorem to find \( \displaystyle \lim_{n \to \infty} \frac{\sin{\left(4 n \right)}}{n^{2}} \).
  1. Since −1 ≤ sin(·) ≤ 1, the expression lies between -1/n**2 and n**(-2).
    Reviewed
  2. \[ \lim_{n \to \infty}\left(- \frac{1}{n^{2}}\right) = 0 \]
    The lower bound's limit.✓ Proved
  3. \[ \lim_{n \to \infty} \frac{1}{n^{2}} = 0 \]
    The upper bound's limit, the same.✓ Proved
  4. Both bounds tend to the same value, so the expression trapped between them does too.
    Reviewed
Answer \( 0 \)

✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 1.630

medium
Use the squeeze theorem to find \( \displaystyle \lim_{n \to \infty} \frac{\cos{\left(2 n \right)}}{n^{2}} \).
  1. Since −1 ≤ cos(·) ≤ 1, the expression lies between -1/n**2 and n**(-2).
    Reviewed
  2. \[ \lim_{n \to \infty}\left(- \frac{1}{n^{2}}\right) = 0 \]
    The lower bound's limit.✓ Proved
  3. \[ \lim_{n \to \infty} \frac{1}{n^{2}} = 0 \]
    The upper bound's limit, the same.✓ Proved
  4. Both bounds tend to the same value, so the expression trapped between them does too.
    Reviewed
Answer \( 0 \)

✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 1.634

medium
Use the squeeze theorem to find \( \displaystyle \lim_{n \to \infty} \frac{\cos{\left(n \right)}}{n} \).
  1. Since −1 ≤ cos(·) ≤ 1, the expression lies between -1/n and 1/n.
    Reviewed
  2. \[ \lim_{n \to \infty}\left(- \frac{1}{n}\right) = 0 \]
    The lower bound's limit.✓ Proved
  3. \[ \lim_{n \to \infty} \frac{1}{n} = 0 \]
    The upper bound's limit, the same.✓ Proved
  4. Both bounds tend to the same value, so the expression trapped between them does too.
    Reviewed
Answer \( 0 \)

✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 1.699

medium
Use the squeeze theorem to find \( \displaystyle \lim_{n \to \infty} \frac{\sin{\left(4 n \right)}}{n} \).
  1. Since −1 ≤ sin(·) ≤ 1, the expression lies between -1/n and 1/n.
    Reviewed
  2. \[ \lim_{n \to \infty}\left(- \frac{1}{n}\right) = 0 \]
    The lower bound's limit.✓ Proved
  3. \[ \lim_{n \to \infty} \frac{1}{n} = 0 \]
    The upper bound's limit, the same.✓ Proved
  4. Both bounds tend to the same value, so the expression trapped between them does too.
    Reviewed
Answer \( 0 \)

✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 1.707

medium
Use the squeeze theorem to find \( \displaystyle \lim_{x \to \infty} 3 + \frac{\cos{\left(x \right)}}{x} \).
  1. Since −1 ≤ cos(·) ≤ 1, the expression lies between 3 - 1/x and 3 + 1/x.
    Reviewed
  2. \[ \lim_{x \to \infty}\left(3 - \frac{1}{x}\right) = 3 \]
    The lower bound's limit.✓ Proved
  3. \[ \lim_{x \to \infty}\left(3 + \frac{1}{x}\right) = 3 \]
    The upper bound's limit, the same.✓ Proved
  4. Both bounds tend to the same value, so the expression trapped between them does too.
    Reviewed
Answer \( 3 \)

✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 1.712

medium
Use the squeeze theorem to find \( \displaystyle \lim_{x \to 0} x \cos{\left(\frac{3}{x} \right)} \).
  1. Since −1 ≤ cos(·) ≤ 1, the expression lies between -Abs(x) and Abs(x).
    Reviewed
  2. \[ \lim_{x \to 0}\left(- \left|{x}\right|\right) = 0 \]
    The lower bound's limit.✓ Proved
  3. \[ \lim_{x \to 0} \left|{x}\right| = 0 \]
    The upper bound's limit, the same.✓ Proved
  4. Both bounds tend to the same value, so the expression trapped between them does too.
    Reviewed
Answer \( 0 \)

✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 1.731

medium
Use the squeeze theorem to find \( \displaystyle \lim_{n \to \infty} \frac{\sin{\left(n \right)}}{n} \).
  1. Since −1 ≤ sin(·) ≤ 1, the expression lies between -1/n and 1/n.
    Reviewed
  2. \[ \lim_{n \to \infty}\left(- \frac{1}{n}\right) = 0 \]
    The lower bound's limit.✓ Proved
  3. \[ \lim_{n \to \infty} \frac{1}{n} = 0 \]
    The upper bound's limit, the same.✓ Proved
  4. Both bounds tend to the same value, so the expression trapped between them does too.
    Reviewed
Answer \( 0 \)

✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 1.739

medium
Use the squeeze theorem to find \( \displaystyle \lim_{x \to \infty} \frac{\sin{\left(x \right)}}{x} \).
  1. Since −1 ≤ sin(·) ≤ 1, the expression lies between -1/x and 1/x.
    Reviewed
  2. \[ \lim_{x \to \infty}\left(- \frac{1}{x}\right) = 0 \]
    The lower bound's limit.✓ Proved
  3. \[ \lim_{x \to \infty} \frac{1}{x} = 0 \]
    The upper bound's limit, the same.✓ Proved
  4. Both bounds tend to the same value, so the expression trapped between them does too.
    Reviewed
Answer \( 0 \)

✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 1.812

medium
Use the squeeze theorem to find \( \displaystyle \lim_{x \to \infty} 3 + \frac{\sin{\left(4 x \right)}}{x} \).
  1. Since −1 ≤ sin(·) ≤ 1, the expression lies between 3 - 1/x and 3 + 1/x.
    Reviewed
  2. \[ \lim_{x \to \infty}\left(3 - \frac{1}{x}\right) = 3 \]
    The lower bound's limit.✓ Proved
  3. \[ \lim_{x \to \infty}\left(3 + \frac{1}{x}\right) = 3 \]
    The upper bound's limit, the same.✓ Proved
  4. Both bounds tend to the same value, so the expression trapped between them does too.
    Reviewed
Answer \( 3 \)

✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 1.815

medium
Use the squeeze theorem to find \( \displaystyle \lim_{x \to \infty} \frac{\cos{\left(x \right)}}{x} \).
  1. Since −1 ≤ cos(·) ≤ 1, the expression lies between -1/x and 1/x.
    Reviewed
  2. \[ \lim_{x \to \infty}\left(- \frac{1}{x}\right) = 0 \]
    The lower bound's limit.✓ Proved
  3. \[ \lim_{x \to \infty} \frac{1}{x} = 0 \]
    The upper bound's limit, the same.✓ Proved
  4. Both bounds tend to the same value, so the expression trapped between them does too.
    Reviewed
Answer \( 0 \)

✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

More the squeeze theorem practice

Use the squeeze theorem to find \( \displaystyle \lim_{x \to \infty} 1 + \frac{\sin{\left(x \right)}}{x} \).
Problem 1.620medium✓ Every equation proved
Use the squeeze theorem to find \( \displaystyle \lim_{x \to \infty} 2 + \frac{\cos{\left(2 x \right)}}{x} \).
Problem 1.622medium✓ Every equation proved
Use the squeeze theorem to find \( \displaystyle \lim_{x \to 0} x^{3} \sin{\left(\frac{4}{x} \right)} \).
Problem 1.625medium✓ Every equation proved
Use the squeeze theorem to find \( \displaystyle \lim_{x \to 0} x^{2} \cos{\left(\frac{4}{x} \right)} \).
Problem 1.631medium✓ Nihil obstat
Use the squeeze theorem to find \( \displaystyle \lim_{x \to 0} x^{2} \cos{\left(\frac{3}{x} \right)} \).
Problem 1.632medium✓ Nihil obstat
Use the squeeze theorem to find \( \displaystyle \lim_{x \to 0} x^{3} \sin{\left(\frac{1}{x} \right)} \).
Problem 1.633medium✓ Every equation proved
Use the squeeze theorem to find \( \displaystyle \lim_{x \to 0} x^{2} \sin{\left(\frac{2}{x} \right)} \).
Problem 1.688medium✓ Nihil obstat
Use the squeeze theorem to find \( \displaystyle \lim_{x \to 0} x^{2} \sin{\left(\frac{3}{x} \right)} \).
Problem 1.689medium✓ Nihil obstat
Use the squeeze theorem to find \( \displaystyle \lim_{x \to 0} x^{3} \sin{\left(\frac{2}{x} \right)} \).
Problem 1.690medium✓ Every equation proved
Use the squeeze theorem to find \( \displaystyle \lim_{x \to \infty} 1 + \frac{\cos{\left(2 x \right)}}{x} \).
Problem 1.691medium✓ Every equation proved
Use the squeeze theorem to find \( \displaystyle \lim_{x \to 0} x^{2} \cos{\left(\frac{1}{x} \right)} \).
Problem 1.692medium✓ Nihil obstat
Use the squeeze theorem to find \( \displaystyle \lim_{x \to \infty} -2 + \frac{\sin{\left(3 x \right)}}{x} \).
Problem 1.693medium✓ Every equation proved
Use the squeeze theorem to find \( \displaystyle \lim_{x \to 0} x \sin{\left(\frac{2}{x} \right)} \).
Problem 1.694medium✓ Nihil obstat
Use the squeeze theorem to find \( \displaystyle \lim_{x \to 0} x^{3} \cos{\left(\frac{3}{x} \right)} \).
Problem 1.695medium✓ Every equation proved
Use the squeeze theorem to find \( \displaystyle \lim_{x \to \infty} -1 + \frac{\cos{\left(3 x \right)}}{x} \).
Problem 1.696medium✓ Nihil obstat
Use the squeeze theorem to find \( \displaystyle \lim_{x \to 0} x^{2} \sin{\left(\frac{1}{x} \right)} \).
Problem 1.697medium✓ Nihil obstat
Use the squeeze theorem to find \( \displaystyle \lim_{x \to \infty} -3 + \frac{\cos{\left(3 x \right)}}{x} \).
Problem 1.698medium✓ Every equation proved
Use the squeeze theorem to find \( \displaystyle \lim_{n \to \infty} \frac{\cos{\left(3 n \right)}}{n^{2}} \).
Problem 1.700medium✓ Nihil obstat
Use the squeeze theorem to find \( \displaystyle \lim_{x \to \infty} -3 + \frac{\cos{\left(2 x \right)}}{x} \).
Problem 1.701medium✓ Every equation proved
Use the squeeze theorem to find \( \displaystyle \lim_{n \to \infty} \frac{\sin{\left(n \right)}}{n^{2}} \).
Problem 1.702medium✓ Every equation proved
Use the squeeze theorem to find \( \displaystyle \lim_{x \to 0} x^{3} \cos{\left(\frac{4}{x} \right)} \).
Problem 1.704medium✓ Every equation proved
Use the squeeze theorem to find \( \displaystyle \lim_{x \to \infty} -3 + \frac{\cos{\left(x \right)}}{x} \).
Problem 1.705medium✓ Every equation proved
Use the squeeze theorem to find \( \displaystyle \lim_{x \to \infty} -2 + \frac{\cos{\left(4 x \right)}}{x} \).
Problem 1.706medium✓ Every equation proved
Use the squeeze theorem to find \( \displaystyle \lim_{x \to 0} x \sin{\left(\frac{1}{x} \right)} \).
Problem 1.708medium✓ Nihil obstat
Use the squeeze theorem to find \( \displaystyle \lim_{x \to \infty} 1 + \frac{\cos{\left(x \right)}}{x} \).
Problem 1.709medium✓ Every equation proved
Use the squeeze theorem to find \( \displaystyle \lim_{x \to \infty} \frac{\cos{\left(2 x \right)}}{x} \).
Problem 1.710medium✓ Nihil obstat
Use the squeeze theorem to find \( \displaystyle \lim_{x \to 0} x \sin{\left(\frac{3}{x} \right)} \).
Problem 1.711medium✓ Nihil obstat
Use the squeeze theorem to find \( \displaystyle \lim_{x \to 0} x \cos{\left(\frac{2}{x} \right)} \).
Problem 1.713medium✓ Nihil obstat
Use the squeeze theorem to find \( \displaystyle \lim_{n \to \infty} \frac{\cos{\left(4 n \right)}}{n} \).
Problem 1.730medium✓ Nihil obstat
Use the squeeze theorem to find \( \displaystyle \lim_{x \to \infty} \frac{\sin{\left(4 x \right)}}{x} \).
Problem 1.732medium✓ Every equation proved
Use the squeeze theorem to find \( \displaystyle \lim_{n \to \infty} \frac{\sin{\left(2 n \right)}}{n^{2}} \).
Problem 1.733medium✓ Nihil obstat
Use the squeeze theorem to find \( \displaystyle \lim_{x \to \infty} \frac{\sin{\left(3 x \right)}}{x} \).
Problem 1.734medium✓ Every equation proved
Use the squeeze theorem to find \( \displaystyle \lim_{x \to 0} x^{3} \sin{\left(\frac{3}{x} \right)} \).
Problem 1.735medium✓ Every equation proved
Use the squeeze theorem to find \( \displaystyle \lim_{x \to \infty} -2 + \frac{\cos{\left(x \right)}}{x} \).
Problem 1.736medium✓ Every equation proved
Use the squeeze theorem to find \( \displaystyle \lim_{x \to \infty} -3 + \frac{\sin{\left(4 x \right)}}{x} \).
Problem 1.737medium✓ Every equation proved
Use the squeeze theorem to find \( \displaystyle \lim_{x \to 0} x^{3} \cos{\left(\frac{2}{x} \right)} \).
Problem 1.738medium✓ Every equation proved
Use the squeeze theorem to find \( \displaystyle \lim_{x \to \infty} 2 + \frac{\cos{\left(4 x \right)}}{x} \).
Problem 1.813medium✓ Every equation proved
Use the squeeze theorem to find \( \displaystyle \lim_{n \to \infty} \frac{\sin{\left(2 n \right)}}{n} \).
Problem 1.814medium✓ Nihil obstat
Use the squeeze theorem to find \( \displaystyle \lim_{x \to \infty} -3 + \frac{\sin{\left(3 x \right)}}{x} \).
Problem 1.816medium✓ Every equation proved
Use the squeeze theorem to find \( \displaystyle \lim_{x \to \infty} 2 + \frac{\sin{\left(3 x \right)}}{x} \).
Problem 1.817medium✓ Every equation proved
Use the squeeze theorem to find \( \displaystyle \lim_{x \to 0} x \cos{\left(\frac{4}{x} \right)} \).
Problem 1.818medium✓ Nihil obstat
Use the squeeze theorem to find \( \displaystyle \lim_{n \to \infty} \frac{\cos{\left(4 n \right)}}{n^{2}} \).
Problem 1.819medium✓ Nihil obstat
Use the squeeze theorem to find \( \displaystyle \lim_{x \to 0} x \cos{\left(\frac{1}{x} \right)} \).
Problem 1.820medium✓ Nihil obstat
Use the squeeze theorem to find \( \displaystyle \lim_{x \to \infty} 3 + \frac{\cos{\left(4 x \right)}}{x} \).
Problem 1.821medium✓ Every equation proved