How Do Limits and Boundedness Apply to Trigonometric Functions?

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SUMMARY

The discussion focuses on the application of limits and boundedness in trigonometric functions, specifically analyzing the inequality x ≤ xsin(1/x) ≤ -x for x < 0 and -|x| ≤ xsin(1/x) ≤ |x| when x is non-zero. Participants confirm that the sine function is bounded, with |sin(a)| ≤ 1, leading to the conclusion that -1 ≤ sin(a) ≤ 1. By substituting a with 1/x and considering the sign of x, the inequalities are correctly manipulated to demonstrate the bounded nature of xsin(1/x).

PREREQUISITES
  • Understanding of trigonometric functions and their properties
  • Knowledge of limits and inequalities in calculus
  • Familiarity with the concept of boundedness in mathematical analysis
  • Basic algebraic manipulation skills
NEXT STEPS
  • Study the properties of the sine function and its boundedness
  • Learn about limits involving trigonometric functions
  • Explore advanced topics in calculus, such as L'Hôpital's Rule
  • Investigate the implications of boundedness in real analysis
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Students of calculus, mathematics educators, and anyone interested in the behavior of trigonometric functions within the context of limits and inequalities.

jason_r
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Show x<(or equal) xsin(1/x) <(or equal) -x if x<0

and -|x| <(or equal) xsin(1/x) <(or equal) |x| if x can't be 0
 
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Well, we know that sine and cosine are bounded right?

so [tex]|sin(a)|\leq 1[/tex] which means that

[tex]-1\leq sina \leq 1[/tex] now in your first part replace a by 1/x than multiply both sides by x, since x is smaller than zero, your inequality sides switch. procede similarly in 2.
 

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