1617-257/Homework Assignment 10

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Contents

Reading

Read, reread and rereread your notes to this point, and make sure that you really, really really, really really really understand everything in them. Do the same every week! Also, read, reread and rereread sections 15-16 of Munkres' book to the same standard of understanding. Remember that reading math isn't like reading a novel! If you read a novel and miss a few details most likely you'll still understand the novel. But if you miss a few details in a math text, often you'll miss everything that follows. So reading math takes reading and rereading and rerereading and a lot of thought about what you've read. Also, preread section 17, just to get a feel for the future.

Doing

Solve all the problems in section 15, but submit only your solutions of problems 1, 3, and 6. In addition, solve and submit your solution of the following problem:

Problem A. Consider the function f\colon[0,\infty)\to{\mathbb R} defined by f(x)=\sin(x^2). Show that

  1. The limit \lim_{x\to\infty}f(x) does not exist.
  2. The limit \lim_{N\to\infty}\int_0^Nf(x)dx does exist (yet do not attempt to compute it).
  3. The extended integral \int_{(0,\infty)}f does not exist.

Submission

Here and everywhere, neatness counts!! You may be brilliant and you may mean just the right things, but if the teaching assistants will be having hard time deciphering your work they will give up and assume it is wrong.

This assignment is due in class on Wednesday December 7 by 2:10PM.

Important

Please write on your assignment the day of the tutorial when you'd like to pick it up once it is marked (Wednesday or Thursday).