Math H161 Brown Au03

Homework 2,  due Wednesday, October 8, 2003

 

Caution: #3 is fairly difficult.  Be sure you read it and start trying it immediately, so that if you don't see how to do it right away it can turn over in your mind for several days before the due date.

 

1.  Textbook, page  73, #1, 2, 3, 4, 5, 6, 7, 10, 11, 13, 14, 15, 16

 

2.  For page 73, #1, prove, using epsilons and deltas, that the function really does converge to the limit you chose.

 

3,   Prove, using epsilons and deltas, that  the limit as x approaches  -1  of the function  f(x) = 10/(3 + x) is 5.  (Sorry,  I can't type the math symbols on this word processor.  This is the same function as p.73, #2,  but  x  is approaching  -1  instead of  2.  Can you think of a pedagogical reason for that change?) Hint:  you'll have to fiddle around like we did in class with the function  x2 + 4 and you'll probably have to finally choose your delta to be the smaller of two numbers.

 

4.  page 305, # 33, 34, 35, 36, 37. You may assume without proof that the limit as  x  approaches 0 of  sinx / x is  1.This is one of the most important limits in calculus.  It is discussed in the text on pages 301-302, although in the text the symbol for the variable is Greek theta  instead of  x.