SUMMARY
The limit of the function f(x) = [x cos(x)] / [x^3 + 1] as x approaches infinity is determined without using L'Hôpital's Rule. The analysis shows that as x tends to infinity, cos(x) oscillates between -1 and 1, while the denominator x^3 + 1 approaches infinity. Thus, the limit simplifies to (cos(x)/x^2) * (1/(1 + 1/x^3)), which approaches 0 as x approaches infinity. Therefore, the limit of f(x) is conclusively 0.
PREREQUISITES
- Understanding of limits in calculus
- Familiarity with trigonometric functions, specifically cosine
- Knowledge of polynomial behavior as x approaches infinity
- Basic understanding of L'Hôpital's Rule and its conditions
NEXT STEPS
- Study the properties of limits involving oscillatory functions
- Learn about the application of L'Hôpital's Rule and its conditions
- Explore advanced limit techniques such as the Squeeze Theorem
- Investigate the behavior of rational functions as x approaches infinity
USEFUL FOR
Students studying calculus, particularly those focusing on limits and trigonometric functions, as well as educators looking for examples of limit evaluation techniques without L'Hôpital's Rule.