SUMMARY
The limit of x^2/(1-cos(x)) as x approaches 0 is determined to be 2 without using L'Hôpital's rule. The solution utilizes the trigonometric identity cos(x) = 1 - 2sin^2(x/2) and the limit property lim_{x→0} (x/sin(x)) = 1. Additionally, multiplying both the numerator and denominator by (1 + cos(x)) simplifies the expression, leading to the conclusion.
PREREQUISITES
- Understanding of trigonometric identities, specifically cos(x) and sin(x).
- Familiarity with limits and continuity in calculus.
- Knowledge of power series expansions, particularly for trigonometric functions.
- Basic algebraic manipulation skills for simplifying expressions.
NEXT STEPS
- Study the derivation and applications of the limit lim_{x→0} (x/sin(x)).
- Explore trigonometric identities and their proofs, focusing on cos(x) and sin(x).
- Learn about power series expansions for various functions, including sin(x) and cos(x).
- Investigate alternative methods for evaluating limits without L'Hôpital's rule.
USEFUL FOR
Students studying calculus, particularly those focusing on limits and trigonometric functions, as well as educators seeking alternative teaching methods for limit evaluation.