SUMMARY
The limit of the expression (x^3 - 6) / (x^k + 3) as x approaches infinity exists when k is equal to 3. By multiplying both the numerator and denominator by 1/x^3, the expression simplifies to 1 / (x^(k-3) + 3/x^3). Analyzing the cases, when k > 3, the limit approaches 0; when k < 3, the limit approaches infinity; and when k = 3, the limit equals 1. Thus, the only value of k that allows the limit to exist is k = 3.
PREREQUISITES
- Understanding of limits in calculus
- Familiarity with polynomial functions
- Knowledge of asymptotic behavior of functions
- Basic algebraic manipulation skills
NEXT STEPS
- Study the concept of limits at infinity in calculus
- Learn about polynomial growth rates and their comparisons
- Explore the epsilon-delta definition of limits
- Investigate L'Hôpital's Rule for indeterminate forms
USEFUL FOR
Students studying calculus, particularly those focusing on limits and asymptotic analysis, as well as educators looking for examples of limit evaluation techniques.