Homework 2.

1. Compute ${lim}_{n\to \mathrm{\infty }}\frac{7n^{18}+5n^5+1}{3n^{18}+5n^3+3}$  (2 points)

2. Compute ${lim}_{n\to \mathrm{\infty }}\sqrt{(1+\frac{1}{n})(1+\frac{1}{n^2})}$  (2 points)

3. Somebody came up with the definition of “bonvergence”: A sequence of real numbers ${\{x_n\}}_{n=1}^{\mathrm{\infty }}$ “bonverges” to a real number $x$ if:

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   • Give an example of a “bonvergent” sequence that is not “convergent”. (justify your answer by giving an explicit $\epsilon$ and $N$) (3 points)

   • Is it true that if ${\{x_n\}}_{n=1}^{\mathrm{\infty }}$ “bonverges” to $x$ and ${\{x_n\}}_{n=1}^{\mathrm{\infty }}$ “bonverges” to $y$, then $x$ always equals to $y$? (justify your answer) (3 points)