Infinite Limit Problems Can some one help me to solve the problems?


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*$$\lim_{n\to\infty}\sum_{k=1}^n\left|e^{2\pi ik/n}-e^{2\pi i(k-1)/n}\right|$$

*$$ \lim_{x\to\infty}\left(\frac{3x-1}{3x+1}\right)^{4x}$$

*$$\lim_{n\to\infty}\left(1-\frac1{n^2}\right)^n $$
(Original scan of problems at http://i.stack.imgur.com/4t12K.jpg)
This questions are from ISI kolkata Computer science Phd exam entrance.The link of the original questions http://www.isical.ac.in/~deanweb/sample/MMA2013.pdf

For the first one, consider a geometric interpretation.  Recall that when $a$ and $b$ are complex numbers, $|a-b|$ is the distance in the plane between the points $a$ and $b$. Peter Tamaroff says in a comment that the limit is $2\pi$, which I believe is correct.
Addendum: The points $e^{2\pi ik/n}$ are spaced evenly around the unit circle.  The distance $\left|e^{2\pi ik/n} - e^{2\pi i(k-1)/n}\right|$ is the distance from one point to the next.  So we are calculating the perimeter of a regular $n$-gon which, as $n$ increases, approximates the perimeter of the unit circle arbitrarily closely.  The answer is therefore $2\pi$. 
For the third one, factor $$\left(1-\frac1{n^2}\right)$$ as 
$$ \left(1-\frac1{n}\right) \left(1+\frac1{n}\right)$$ and then apply the usual theorem about $\lim_{n\to\infty}\left(1+\frac an\right)^n$.
