Limit of a product of sin and a polynomial

[smashX]

Homework Statement

Given a n-sided polygon. Divide it into multiple small triangles that have same radius r. Compute the following limits:

Homework Equations

The Attempt at a Solution

When I plug in a random number to guess the answer, somehow it is always near 0 (very small) so my guess is 0. Having said that, I'm still stuck at how to compute this limit and I really want to ask you guys for advice. Any suggestions is appreciated, thanks.

 

[hunt_mat]

From the construction it is going to tend to the area of a circle which is pi*r^2. For this you will need the limit:
[tex]
\lim_{x\rightarrow 0}\frac{\sin x}{x}=1
[/tex]
So you will need to manipulate your limit. Hint, if n goes to infinite what does 1/n tend to?

 

[SammyS]

Use L'Hopital's rule.[tex]\frac{1}{2}nr^2\sin\left(\frac{2\pi}{n}\right)= \frac{1}{2}r^2\frac{\sin\left(\frac{2\pi}{n}\right)}{n}[/tex]is of the form 0/0.

 

[Mark44]

Use L'Hopital's rule.[tex]\frac{1}{2}nr^2\sin\left(\frac{2\pi}{n}\right)= \frac{1}{2}r^2\frac{\sin\left(\frac{2\pi}{n}\right)}{n}[/tex]is of the form 0/0.

That should be
[tex]\frac{1}{2}nr^2\sin\left(\frac{2\pi}{n}\right)= \frac{1}{2}r^2\frac{\sin\left(\frac{2\pi}{n}\right)}{1/n}[/tex]