SUMMARY
The limit of the expression \(\lim_{n\to\infty} \frac{\sin(nt)}{n}=0\) for \(t \in [0,1]\) is established through the properties of bounded functions and the behavior of sequences. Since the sine function is bounded by -1 and 1, the fraction \(\frac{\sin(nt)}{n}\) approaches zero as \(n\) increases indefinitely. This conclusion is derived from the fact that the numerator remains bounded while the denominator grows without bound.
PREREQUISITES
- Understanding of limits in calculus
- Familiarity with the properties of the sine function
- Basic knowledge of sequences and series
- Concept of bounded functions
NEXT STEPS
- Review the definition of limits in calculus
- Study the properties of bounded functions and their implications
- Explore examples of limits involving trigonometric functions
- Learn about sequences and their convergence criteria
USEFUL FOR
Students of calculus, educators teaching limit concepts, and anyone looking to refresh their understanding of trigonometric limits.