SUMMARY
The limit problem presented is to find \(\lim_{x \to 0}\frac{1-\cos(x)\cos(2x)\cos(3x)}{1-\cos(x)}\). The solution involves rewriting the numerator using the identity \(\lim_{x \to 0}\frac{1-\cos(x)+\cos(x)[1-\cos(2x)+\cos(2x)(1-\cos(3x)]}{1-\cos(x)}\). This transformation simplifies the limit and reveals the underlying relationships between the cosine functions. The key insight is recognizing the addition and subtraction of \(\cos(x)\) in the numerator as a critical step in solving the limit.
PREREQUISITES
- Understanding of trigonometric limits
- Familiarity with the cosine function and its properties
- Knowledge of limit laws and algebraic manipulation
- Ability to expand trigonometric identities
NEXT STEPS
- Study the derivation of trigonometric limits using Taylor series expansions
- Learn about L'Hôpital's Rule for evaluating indeterminate forms
- Explore the properties of cosine functions and their behavior near zero
- Practice solving similar limit problems involving trigonometric functions
USEFUL FOR
Students studying calculus, particularly those focusing on limits and trigonometric functions, as well as educators looking for examples of limit evaluation techniques.