SOLUTION: PROVING A. limit as delta x approaches to zero cos delta x-1/delta x B. limit as delta x approaches to zero sin delta x/delta x

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Question 1131261: PROVING
A. limit as delta x approaches to zero cos delta x-1/delta x
B. limit as delta x approaches to zero sin delta x/delta x
Answer by solver91311(24713) About Me  (Show Source):
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Segment SC = by definition (note unit circle). Altitude of triangle OSU is and the base of triangle OSU is 1, so the area of triangle OSU is .

The area of a circle sector is given by where is the subtended angle. Hence, the area of sector OSU is simply

Segment TU = by definition of the tangent function. (TU divided by OU, but OU measures 1). Then the area of triangle OTU is .

From the diagram, the area of triangle OSU is less than or equal to the area of sector OSU which in turn is less than or equal to triangle OTU. Hence:



Multiply by



Take the reciprocal and reverse the sense of the inequalities:



But and

Therefore, by the Squeeze Theorem:




John

My calculator said it, I believe it, that settles it