For 13 minutes (21:00-34:30), Professor Jerison tries to justify (1-cos x)/x goes to zero, without, I think, making the case. While he convinces me that the "1-cosx gap" goes to zero faster than the arc length, he doesn't show that it goes to zero fast enough. For example, if it went to zero 10 times faster than the arc length, the limit would approach 1/10, not zero. I'm comfortable with his argument about (sinx)/x going to 1 and have in fact used that argument to defend the "small angle" approximation in Physics. (Larson 3rd edition, pg. 81 however, gives a rigorous explanation using limiting cases of  triangular areas bounding the section of the circular area swept out by the angle, and then applying the squeeze theorem, (1-cosx)/x is left as an exercise. But once you have the limit of (sinx)/x going to 1, just multiply top and bottom of (1-cosx)/x by (1+cos), the numerator becomes (sinx)^2  and the expression becomes: [(sinx)/x] [(sinx)/(1+cosx)], first factor goes to 1 and the second to zero. It's good to review properties of limits for this.Where fault can be found, the good is often ignored. MIT and the professors involved are doing a GREAT public service by constructing these online courses. I'm amazed at the breadth and complexity of the courses offered. I deeply thank all of the MIT people involved in this. The examples and problems done in the class are excellent. I find many of the proofs easier to follow than in some texts. Thanks Professor Jerison!.####[]