SUMMARY
The limit of the sequence defined by xn = n(sqrt(n+1) - sqrt(n))^2 as n approaches infinity is computed by first expanding the expression. The binomial expansion leads to the form (2n + 1) - {2 · sqrt(n+1) · sqrt(n)}, which remains indeterminate. To resolve this, multiplying by the conjugate (2n + 1) + {2 · sqrt(n+1) · sqrt(n)} allows simplification. Ultimately, applying the limit of a rational function reveals the behavior as n approaches infinity.
PREREQUISITES
- Understanding of limits in calculus
- Familiarity with binomial expansion
- Knowledge of rational functions and their limits
- Experience with algebraic manipulation of expressions
NEXT STEPS
- Study the properties of limits in calculus
- Learn about binomial expansion techniques
- Explore rational functions and their asymptotic behavior
- Practice solving limits involving square roots and conjugates
USEFUL FOR
Students studying calculus, particularly those focusing on limits and algebraic manipulation, as well as educators looking for examples of limit computation techniques.
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