anonymous
  • anonymous
Hello, In video lecture 3, Prof. Jerisson proved that limit of (1- cos theta)/theta as theta approaches 0 is 0. I understand why (1- cos theta) tend to 0. But what about theta itself in the denominator? As questioned by one student there, why didn't it become 0/0? I'm still confused. Anybody can explain further please?
OCW Scholar - Single Variable Calculus
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jamiebookeater
  • jamiebookeater
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phi
  • phi
Yes, the explanation was very "hand-wavy" The short answer is if you have a ratio of two things that "go to 0" *but* the top value goes to zero *faster*, then the ratio approaches zero. for example \( \frac{x^2}{x} \) when x=0.1, x^2 =0.01. when x=0.01, x^2= 0.0001, etc Of course we could simplify that to just x, and it evidently goes to zero... but the idea is still valid... the top and bottom can go to zero at different rates, and it is a race which wins.
phi
  • phi
For a specific argument see the attached file. For a clear explanation of the limit of sin x/ x you might want to see https://www.khanacademy.org/math/differential-calculus/limits_topic/squeeze_theorem/v/proof-lim-sin-x-x and for better background on limits, see http://ocw.mit.edu/resources/res-18-006-calculus-revisited-single-variable-calculus-fall-2010/part-i-sets-functions-and-limits/lecture-5-a-more-rigorous-approach-to-limits/
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