SUMMARY
The limit $$\lim_{{x}\to{\infty}} \sqrt{x^2 + 2x + 1} - x$$ evaluates to 1. This conclusion is reached by rewriting the expression under the square root as a perfect square, specifically $$(x + 1)^2$$. The calculation shows that as x approaches infinity, the expression simplifies correctly to yield the limit of 1, contrary to the initial assumption of 0.
PREREQUISITES
- Understanding of limits in calculus
- Familiarity with algebraic manipulation of expressions
- Knowledge of square roots and their properties
- Concept of infinity in mathematical analysis
NEXT STEPS
- Study the properties of limits involving infinity
- Learn about L'Hôpital's Rule for indeterminate forms
- Explore algebraic techniques for simplifying radical expressions
- Investigate the concept of asymptotic behavior in calculus
USEFUL FOR
Students of calculus, mathematics educators, and anyone interested in understanding limits and their applications in mathematical analysis.