The limit as x→0 of sinx/x

Author:
Ryan Hirst
GeoGebra Applet
Two limits: 1) PQ approaches the arc s: The arc s is caught between PQ and PC +CQ < PB. Drive θ to zero. The differences between the three lengths can be made as small as we wish (smaller than any finite quantity). Result 1: In the limit, the tangent PB, arc s, and subtended chord PQ are equal. We may use them interchangeably. 2) lim sinθ/θ = 1:
  • Put sinθ and θ in a box together. Pick θ or sinθ. I choose θ. Create two inequalities: θ > A sinθ θ < B sinθ A, B expressions in θ and x. This is the box: A sinθ < θ < B sinθ Put sinθ/θ in the middle.
  • Shut the box: drive x to 0 (the limiting value).
  • The left and right sides are defined and positive for all values of x < 1. How far apart are they at x=0? They are the same.
It is always possible to take a limit in this way (place the ratio in question at the center of an inequality, and let the bounds approach the limit). The upside and downside are the same: I will get the limiting interval I asked for. I could choose badly and get an answer like: -∞ <sinθ/θ < ∞. Here, I found two expressions which are equal at θ=0.

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