114 Math Fall 2005 Lab 3
We can modify the definitions on page 119 in various ways. For example: The definition of the assertion
xa+
lim f (x) =
is that for every positive number B there is a positive such that f (x) will exceed B whenever x is between a and a + . Here is a picture for this definition. Note y > B when x is between a and a + .
B
y=f(x) a d
By making similar modifications, we can come up with definitions for the following assertions: lim f (x) = lim f (x) = - lim f (x) = - lim f (x) = lim f (x) = L lim f (x) = - lim f (x) = L
xa-
xa+
xa-
x
x-
x
lim f (x) =
x
lim f (x) = -
x-
x-
If we consider f (x) =
verify this with Maple by limit(3/(x-1),x=1,right); We could obtain a plot with Maple by plot(3/(x-1),x=0..2,y=-50..50,discont=true);
3 then we know that lim f (x) = . We can x1+ x-1
40
20
y
0 0 0.5 1 x -20 1.5 2
-40
In view of the definition of the limit, it is fair to ask this quantitative question: How close does x need to be to 1 to be sure that f (x) will be than greater 10,000? This amounts to solving the inequality f (x) > 10, 000. We can solve this with Maple as follows: solve(3/(x-1)>10000); solve(3/(x-1)>10000.); We see that = 0.003 is sufficiently small.
Lab Report (due Sept. 29)
Names: Lab Time: For each of the following limits carefully state the definition and provide a sketch, as described on the first page. Then answer the quantitative question. Use Maple as a tool as you see fit. Explain how you come to your conclusion. 1. lim+ f (x) = -
xa
Since lim +
8 = -, how close to 2 does x need to be to be sure x2 2 - x that f (x) will be less...