1. Show lim Θ/sinΘ=1

Θ->0

(Hint: show sinΘcosΘ <Θ<tanΘ)

I have no idea to do this problem, other than use the squeeze theorem, but i still dont know what to do. any help would be greatly appreciated.

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- Oct 4th 2008, 07:33 AMjohntuanlimits and continuity
1. Show lim Θ/sinΘ=1

Θ->0

(Hint: show sinΘcosΘ <Θ<tanΘ)

I have no idea to do this problem, other than use the squeeze theorem, but i still dont know what to do. any help would be greatly appreciated. - Oct 4th 2008, 08:05 AMskeeter
here is a link to the classical geometric approach in determining this limit ...

http://www.csun.edu/~ac53971/courses/math350/xtra_sine.pdf - Oct 4th 2008, 08:32 AMbkarpuz
I will show my own proof for this.

**Proof**.

First draw a unit circle, and put a equilateral -polygon in it.

Then from the center of the circle, draw lines to the corners of the polygon.

We see that the center angle is divided to , thus all the triangles have as the vertex of the isosceles triangle, and each triangles have the area .

Hence the sum of the areas of the triangles is .

We can see that letting tend to infinity tends to the are a of the circle .

That is

or equivalently

.

Substitute , then we see that as , which yields

I hope this is helpful. (Wink)

**Note**. Here, travels through rational numbers and since means accumulation, we may think that travels through reals since it is the closure of rational numbers.