Proof of Trig Limits

Proof of Trig Limits - Proof of Trig Limits In this section...

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Proof of Trig Limits In this section we’re going to provide the proof of the two limits that are used in the derivation of the derivative of sine and cosine in the Derivatives of Trig Functions section of the Derivatives chapter. Proof of : This proof of this limit uses the Squeeze Theorem . However, getting things set up to use the Squeeze Theorem can be a somewhat complex geometric argument that can be difficult to follow so we’ll try to take it fairly slow. Let’s start by assuming that . Since we are proving a limit that has it’s okay to assume that is not too large ( i.e. ). Also, by assuming that is positive we’re actually going to first prove that the above limit is true if it is the right-hand limit. As you’ll see if we can prove this then the proof of the limit will be easy. So, now that we’ve got our assumption on taken care of let’s start off with the unit circle circumscribed by an octagon with a small slice marked out as shown below. Points A and C are the midpoints of their respective sides on the octagon and are in fact tangent to the circle at that point. We’ll call the point where these two sides meet

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