SUMMARY
The limit of the function \(\lim_{x\rightarrow 0} \frac{\sin(3x)}{\sin(5x)}\) is established as 0.6. The solution involves recognizing the continuity of the sine function and applying the limit property \(\lim_{u \to 0} \frac{\sin u}{u} = 1\). By strategically multiplying the expression by \(\frac{3x}{3x}\) and \(\frac{5x}{5x}\), the limit can be simplified to yield the correct result. This method is crucial for proving limits involving trigonometric functions.
PREREQUISITES
- Understanding of trigonometric limits, specifically \(\lim_{u \to 0} \frac{\sin u}{u}\)
- Basic knowledge of continuity in calculus
- Familiarity with limit notation and evaluation techniques
- Ability to manipulate algebraic expressions involving limits
NEXT STEPS
- Study the properties of limits in calculus, focusing on trigonometric functions
- Explore advanced limit techniques, such as L'Hôpital's Rule
- Learn about continuity and differentiability in calculus
- Practice solving similar limit problems involving sine and cosine functions
USEFUL FOR
Students studying calculus, particularly those focusing on limits and trigonometric functions, as well as educators seeking to enhance their teaching methods in these topics.