TEST 2 (MAT 370, REVIEW)

ASU ID:..................................NAME:..................................

1. Establish whether the set is open. Justify your answer.

Set 1:

$$\cup_{n=1}^\infty \{\frac{1}{n} \le x \le 3 - \frac{1}{n} \}$$

Set 2:

$$\cup_{n=1}^\infty \{\frac{1}{n}\le x \le 3 + \frac{1}{n} \}$$

2. Find all limit points for the following set.

$$\{\frac{(-1)^n n + 3}{3n + 1};\;n=1,\;2,\;\dots \}$$

3. Establish whether $\frac{1}{4}$ is a limit point of Cantor set.

4. State whether Cantor set without point $\frac{1}{4}$ is compact. Prove your statement.

5. Find an approximation of each of the zeroes for function

$$x^4 + x -1$$

Your approximation $r$ should satisfy

$$\mid r^4 + r -1 \mid < 0.01$$

6. How one can define each of the following functions at a given point $x_0$ so that the resulted function is continuous at $x_0?$

Function 1:

$$f(x)= \frac{x^2 -1 }{x+1}\;\;\mbox{ and } x_0 = -1$$

Function 2:

$$f(x)=\frac{x^2+3x+2}{x+2} \;\;\mbox{ and } x_0 = -2$$