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

To solve the first one, note $$\begin{align}
  \mathop {\lim }\limits_{n \to \infty } \sum\limits_{k = 1}^n {\left| {{e^{2\pi ik/n}} - {e^{2\pi i\left( {k - 1} \right)/n}}} \right|}  &= \mathop {\lim }\limits_{n \to \infty } \sum\limits_{k = 1}^n {\left| {{e^{2\pi ik/n}} - \frac{{{e^{2\pi ik/n}}}}{{{e^{2\pi i/n}}}}} \right|}  \cr 
   &= \mathop {\lim }\limits_{n \to \infty } \sum\limits_{k = 1}^n {\left| {1 - \frac{1}{{{e^{2\pi i/n}}}}} \right|}  \cr 
   &= \mathop {\lim }\limits_{n \to \infty } \sum\limits_{k = 1}^n {\left| {\frac{{{e^{2\pi i/n}} - 1}}{{{e^{2\pi i/n}}}}} \right|}  \cr 
   &= \mathop {\lim }\limits_{n \to \infty } \sum\limits_{k = 1}^n {\left| {{e^{2\pi i/n}} - 1} \right|}  \cr 
   &= \mathop {\lim }\limits_{n \to \infty } n\left| {{e^{2\pi i/n}} - 1} \right| \cr 
   &= \left| {\mathop {\lim }\limits_{n \to \infty } \frac{{{e^{2\pi i/n}} - 1}}{{1/n}}} \right| \cr 
   &= \left| {\frac{d}{{dx}}{{\left. {{e^{2\pi ix}}} \right|}_{x = 0}}} \right| \cr 
   &= \left| {2\pi i} \right| \cr 
  & = 2\pi  \end{align}$$
