**Math 1572H Honors Calculus Spring Semester 2002**

Assignment 7 - Due Thursday 3/14/2002

Hand in the starred (*) problems.

**Read:** Simmons, Sections 13.3, 13.4. I was over-enthusiastic in the last assignment in expecting to get to section 13.3, so the questions on that section from assignment 6 are **not **due on 3/7/02, but reappear here. This is difficult material, so I only specify two sections.

Exercises:

Sec. 13.3: 2b, 2i, 2l*, 5b, 5e*, 5f, 5g*, 5h, 10a, 12*

Sec. 13.4: 1, 2, 2a*, 2b*, 2c*, 2e* 3c, 3d, 6, 8*

A*. (This in an exercise on the logic of the definition of the limit of a sequence given on page 433.) Consider whether or not a sequence {x_{n}} has the number 7 as its limit. Which of the following correctly expresses the statement that x_{n} does **not** tend to 7 as n tends to infinity. Give some brief explanation!

(i) For each positive number e and for every positive integer n_{0} there exists a number n > n_{0} with |x_{n} - 7| > e.

(ii) There is a positive number e for which there is a positive integer n_{0} so that for every number n > n_{0} we have |x_{n} - 7| > e.

(iii) There is a positive number e such that for every positive integer n_{0} there is a number n > n_{0} with |x_{n} - 7| > e.

B*. Let . Find a number n_{0} so that for every n > n_{0} we have a_{n} < 1/10.