SUMMARY
The limit function f(x) = lim _{n->\infty}(x{n})/(1+x{n}) evaluates to 1 when x > 1, 0 when 0 < x < 1, and 1/2 when x = 1. The reasoning behind this conclusion is that as n approaches infinity, the highest power of n in both the numerator and denominator dominates, leading to the simplification x^n/x^n = 1. Therefore, the correct interpretation of the limit is crucial for understanding the behavior of the function as n approaches infinity.
PREREQUISITES
- Understanding of limits in calculus
- Familiarity with polynomial functions
- Knowledge of asymptotic behavior
- Basic algebraic manipulation skills
NEXT STEPS
- Study the concept of limits in calculus, focusing on infinity behavior
- Explore polynomial functions and their properties
- Learn about asymptotic analysis in mathematical functions
- Practice algebraic manipulation of rational functions
USEFUL FOR
Students studying calculus, educators teaching limit concepts, and anyone interested in the behavior of functions as they approach infinity.