SUMMARY
The limit of the expression lim (x approach infinity) [(sqrt(x+1)-1)/x] can be evaluated by multiplying the numerator and denominator by the conjugate (sqrt(x + 1) + 1). This simplifies the expression to lim (x approach infinity) [(x + 1 - 1)/(x(sqrt(x + 1) + 1))], which further reduces to lim (x approach infinity) [1/(sqrt(x + 1) + 1)]. As x approaches infinity, the limit converges to 0.
PREREQUISITES
- Understanding of limits in calculus
- Familiarity with algebraic manipulation of expressions
- Knowledge of conjugates in simplifying expressions
- Basic proficiency in evaluating limits at infinity
NEXT STEPS
- Study the properties of limits in calculus
- Learn about the use of conjugates in algebraic simplification
- Explore advanced limit techniques such as L'Hôpital's Rule
- Practice evaluating limits involving square roots and rational expressions
USEFUL FOR
Students studying calculus, particularly those focusing on limits, algebraic manipulation, and simplification techniques.