August 23 - Limit sin x / x as x->0

We spent some time talking about the true/false problems from the homework on Friday. I think that the issue there is getting comfortable with If –, then – statements perhaps more than the actual limits concepts involved.

The main topic for today was the limit as x->0 of sin(x) / x. See the diagram below:


Angle theta is drawn in standard position on the unit circle. Thus the dashed red line has length sin(theta) (since the hypotenuse is 1) and the thick green arc has length theta (Arc Length = Radius * Central Angle and the radius = 1). Clearly the arc has greater length than the triangle side, but as the angle theta gets smaller, the side of the triangle and the length of the arc get closer to being the same length. Or alternatively, if their lengths are growing similar, the ratio of the lengths is approaching 1. Thus an argument that as theta->0, sin(theta) / theta -> 1

We worked some examples to illustrate how to work problems with trig limits. Those are copied below: