Solutions to Problem Set 18

Math 310-01/02

10-23-2017

1. Prove that $\displaystyle
   \lim_{n \to \infty} \dfrac{2 n}{n + 1} = 2$ .

Let $\epsilon > 0$ . Set $M = \max\left(0, \dfrac{2}{\epsilon} - 1\right)$ . If $n
   > M$ , then

$$n > M \ge 0 \quad\hbox{and}\quad n > M \ge \dfrac{2}{\epsilon} - 1.$$

So

$$\eqalign{ n & > \dfrac{2}{\epsilon} - 1 \cr n + 1 & > \dfrac{2}{\epsilon} \cr \epsilon & > \dfrac{2}{n + 1} \cr \epsilon & > \left|\dfrac{2}{n + 1}\right| \cr \epsilon & > \left|\dfrac{-2}{n + 1}\right| \cr \epsilon & > \left|\dfrac{2 n - 2(n + 1)}{n + 1}\right| \cr \epsilon & > \left|\dfrac{2 n}{n + 1} - 2\right| \cr}$$

In the third step, the inequality is not reversed on division by $n + 1$ , because $n > M
   \ge 0$ implies $n + 1 > 1 > 0$ .

This proves that $\displaystyle
   \lim_{n \to \infty} \dfrac{2 n}{n + 1} = 2$ .


2. Use the limit definition to prove that

$$\lim_{n \to \infty} \dfrac{12 n + 2}{3 n + 1} = 4.$$

Let $\epsilon > 0$ . Set $M = \max \left(0, \dfrac{2 - \epsilon}{3\epsilon}\right)$ . Then

$$M \ge 0 \quad\hbox{and}\quad M \ge \dfrac{2 - \epsilon}{3\epsilon}.$$

Suppose $n > M$ . Then

$$n > 0 \quad\hbox{and}\quad n > \dfrac{2 - \epsilon}{3\epsilon}.$$

Using the second inequality,

$$\eqalign{ n & > \dfrac{2 - \epsilon}{3\epsilon} \cr 3 n \epsilon & > 2 - \epsilon \cr 3 n \epsilon + \epsilon & > 2 \cr \epsilon(3 n + 1) & > 2 \cr \noalign{\vskip2pt} \epsilon & > \dfrac{2}{3 n + 1} \cr}$$

In the last step, the inequality is not reversed, since $n > 0$ implies $3 n + 1
   > 1 > 0$ . The latter implies that $\dfrac{2}{3 n + 1} > 0$ , so

$$\eqalign{ \epsilon & > \left|\dfrac{2}{3 n + 1}\right| \cr \noalign{\vskip2pt} \epsilon & > \left|\dfrac{-2}{3 n + 1}\right| \cr \noalign{\vskip2pt} \epsilon & > \left|\dfrac{12 n + 2 - 4(3 n + 1)}{3 n + 1}\right| \cr \noalign{\vskip2pt} \epsilon & > \left|\dfrac{12 n + 2}{3 n + 1} - 4\right| \cr}$$

This proves that $\displaystyle
   \lim_{n \to \infty} \dfrac{12 n + 2}{3 n + 1} = 4$ .


The earth keeps some vibrations going There in your heart, and that is you. - Edgar Lee Masters


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