SUMMARY
The limit of sin|x|/x as x approaches 0 is determined by evaluating the one-sided limits. The left-hand limit, lim x→0-, results in -1, while the right-hand limit, lim x→0+, results in 1. Since these two one-sided limits are not equal, the overall limit does not exist. This conclusion is reached by analyzing the behavior of the absolute value function in conjunction with the sine function near zero.
PREREQUISITES
- Understanding of limits in calculus
- Familiarity with one-sided limits
- Knowledge of the sine function and its properties
- Basic grasp of absolute value functions
NEXT STEPS
- Study the concept of one-sided limits in depth
- Explore the properties of the sine function, particularly near zero
- Learn about the epsilon-delta definition of limits
- Investigate the implications of discontinuities in functions
USEFUL FOR
Students studying calculus, particularly those focusing on limits and continuity, as well as educators seeking to explain the nuances of limit evaluation involving absolute values and trigonometric functions.