SUMMARY
The limit of the function f(x) = x^2 ln(1+1/x) - x as x approaches infinity is determined to be -1/2. The solution involves applying L'Hôpital's Rule twice after transforming the function into f(u) = (ln(1+u) - u)/u^2, where u = 1/x. The derivative calculations confirm that the second term approaches -1, leading to the conclusion that the overall limit is indeed -1/2.
PREREQUISITES
- Understanding of limits in calculus
- Familiarity with L'Hôpital's Rule
- Knowledge of logarithmic functions
- Basic differentiation techniques
NEXT STEPS
- Study advanced applications of L'Hôpital's Rule in limit problems
- Explore the properties of logarithmic functions and their limits
- Learn about Taylor series expansions for approximating functions
- Investigate the behavior of functions as they approach infinity
USEFUL FOR
Students studying calculus, particularly those focusing on limits and derivatives, as well as educators looking for examples of applying L'Hôpital's Rule in limit evaluations.