Homework #13
Limits at a Real Number
Read section 3.2
Exercises from the text: Pages 131-132: #1a,c,f, #5, #6, #8, #13-15
Other exercises in thought and logic that you will most certainly find interesting.
1.
 x if x is rational
Let f ( x)  
. Determine, with proof
1  x if x is irrational
a. Lim f ( x)
x 0
b. Lim f ( x)
x
1
2
The following questions involve the concept of a monotone function.
Definition: A function f : D  is said to be monotone increasing [decreasing] if and
only if for each x, y  D with x  y, f ( x)  f ( y ) [ f ( x)  f ( y )] .
Prove each of the following:
2.
Suppose that f :[0,1]  is monotone increasing. Then the set
D  {x  [0,1] : f does not have a limit at x} is countable. Further, if f has a limit at
c  (0,1) , then Lim f ( x)  f (c) .
x c
3.
If f :[0,1] 
is monotone, then both Lim f ( x) and Lim f ( x) exist.
x 0
x 1