Proving the limit of sinx/x as x approaches 0 equals 1

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SUMMARY

The limit of sin(x)/x as x approaches 0 equals 1 is proven using geometric principles outlined in James Stewart's calculus textbook, specifically on page 192. The discussion highlights a sequence of inequalities involving the arc length and straight segments, clarifying the transition from EB to ED. The proof utilizes trigonometric functions, particularly the relationship tan(θ) = |AD|/|OA|, to simplify the expression. This geometric approach effectively demonstrates the limit through established inequalities.

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Terrell
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this is a geometric proof from James Stewart's calculus textbook page 192. I'm confused in the sequence of inequalities as part of the proof...
theta = arcAB < AB + EB ==> arcAB < AE + ED. How did EB turned into ED?
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In the first diagram, it' easy to see that ##\left|EB\right| < \left|ED\right|##, so what you ask follows. Then there are just substitutions.
 
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Point D was described in the first half of the proof. It should be clear that |ED| > |EB|.
The benefit of doing the substitution is that you can reduce the expression to one involving straight segments related to theta by trigonometric functions.
##\tan \theta = \frac{|AD|}{|OA|}##
 
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