Math 2.0

Special functions Proposition

sin x / x tends to 1

Builds on Derivative of sineDerivativePower series of sine

Statement

lim⁡x→0sin⁡xx=1.\lim_{x \to 0} \frac{\sin x}{x} = 1.

Remarks

The quotient is the difference quotient of sin⁡\sin at 00. So the limit is sin⁡′(0)=cos⁡0=1\sin'(0) = \cos 0 = 1, by the derivative of sine. The series gives sin⁡xx=1−x26+⋯\frac{\sin x}{x} = 1 - \frac{x^2}{6} + \cdots for x≠0x \ne 0. A geometric proof uses cos⁡x≤sin⁡xx≤1\cos x \le \frac{\sin x}{x} \le 1 for 0<∣x∣<π/20 < |x| < \pi/2 and the squeeze theorem. Mathlib states it as Real.tendsto_sin_div_nhdsNE_zero.

Reading path

To understand this page from the ground up, read these 66 pages in order. The foundations come first.

  1. Limit point of a set definition
  2. Local maximum definition
  3. Sequence definition
  4. Limit of a function definition
  5. Local minimum definition
  6. Real sequence definition
  7. Subsequence definition
  8. Continuity at a point definition
  9. Derivative definition
  10. Bounded sequence definition
  11. Cauchy sequence definition
  12. Geometric sequence definition
  13. Limit of a sequence definition
  14. Monotone sequence definition
  15. Series definition
  16. Continuous function definition
  17. Differentiable function definition
  18. Fermat's theorem on stationary points statement
  19. Limit superior definition
  20. Cauchy sequences are bounded statement
  21. Sum of a finite geometric series statement
  22. Monotone convergence theorem for sequences statement
  23. Every sequence has a monotone subsequence statement
  24. Convergent sequence definition
  25. Pointwise convergence definition
  26. Squeeze theorem statement
  27. Partial sum definition
  28. Power series definition
  29. Derivative function definition
  30. Bolzano–Weierstrass theorem statement
  31. The sequence r^n tends to zero for |r| < 1 example
  32. Radius of convergence definition
  33. Convergent series definition
  34. Uniform convergence definition
  35. Sine and cosine definition
  36. Higher derivative definition
  37. Boundedness theorem statement
  38. Cauchy criterion for sequences statement
  39. Uniform convergence on a set definition
  40. Absolute convergence definition
  41. A series of nonnegative terms converges iff its partial sums are bounded statement
  42. Series of functions definition
  43. Sum of a series definition
  44. Taylor polynomial definition
  45. Continuously differentiable function definition
  46. Extreme value theorem statement
  47. Uniformly Cauchy sequence of functions definition
  48. Geometric series example
  49. Comparison test statement
  50. Smooth function definition
  51. Rolle's theorem statement
  52. Root test statement
  53. Ratio test statement
  54. Weierstrass M-test statement
  55. Mean value theorem statement
  56. Cauchy–Hadamard theorem statement
  57. Ratio formula for the radius of convergence statement
  58. Taylor series definition
  59. Power series converge uniformly on smaller discs inside the radius of convergence statement
  60. Uniform convergence of derivatives allows term-by-term differentiation statement
  61. Maclaurin series definition
  62. Power series of sine statement
  63. Term-by-term differentiation statement
  64. Power series are differentiable inside the radius of convergence statement
  65. Derivative of sine statement
  66. sin x / x tends to 1 statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Differentiation_of_trigonometric_functions, Limit of sin(θ)/θ as θ tends to 0
  2. Web https://en.wikipedia.org/wiki/Sinc_function, Properties
About this page
Tag01QSSKS5C1
URIm2://01QSSKS5C1
Kindstatement, proposition
WritingW0 stub
MSC33B10
StatusThis copy of the site has no build data. Run m2 status 01QSSKS5C1.
UsesDerivative of sine, Derivative, Power series of sine

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