SUMMARY
The limit of the function sin(x)/(x + sin(x)) as x approaches 0 can be evaluated without using L'Hôpital's rule by applying Taylor series expansion. The Taylor series for sin(x) around x = 0 is sin(x) = x - x^3/6 + O(x^5). By substituting this into the limit expression and simplifying, one can find the limit directly. Dividing both the numerator and denominator by x also aids in resolving the limit as x approaches 0.
PREREQUISITES
- Understanding of limits in calculus
- Familiarity with Taylor series expansion
- Basic knowledge of trigonometric functions
- Ability to manipulate algebraic expressions
NEXT STEPS
- Study the Taylor series for common functions, particularly sin(x) and cos(x)
- Learn techniques for evaluating limits without L'Hôpital's rule
- Practice manipulating limits involving trigonometric functions
- Explore alternative methods for limit evaluation, such as the epsilon-delta definition
USEFUL FOR
Students in calculus courses, educators teaching limit concepts, and anyone seeking to deepen their understanding of trigonometric limits without relying on L'Hôpital's rule.