Thread: limits

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.

here is a link to the classical geometric approach in determining this limit ...

http://www.csun.edu/~ac53971/courses/math350/xtra_sine.pdf

Originally Posted by johntuan

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.

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.
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.