SUMMARY
The sequence defined by a(n) = (n + 7) / (2 + sin(n)) tends to infinity as n approaches infinity. L'Hôpital's Rule is not applicable in this case since the denominator does not approach zero or infinity. Instead, by establishing that 1 ≤ 2 + sin(n) ≤ 3 for all n, one can derive both lower and upper bounds for the sequence. As n becomes large, both bounds converge to infinity, confirming that a(n) indeed tends to infinity.
PREREQUISITES
- Understanding of limits in calculus
- Familiarity with L'Hôpital's Rule
- Knowledge of bounded functions and their properties
- Basic trigonometric functions and their behavior
NEXT STEPS
- Study the application of L'Hôpital's Rule in different scenarios
- Explore the concept of bounding sequences in calculus
- Learn about the behavior of trigonometric functions over intervals
- Investigate the formal definition of limits and their proofs
USEFUL FOR
Students studying calculus, particularly those focusing on sequences and limits, as well as educators looking for examples of proving the behavior of sequences.