Section 1.2
25
Section 1.2 Introduction to Limits
description
Section 1.2. Introduction to Limits. Definition of Limit. If f ( x ) becomes arbitrarily close to a unique number L as x approaches c from either side, the limit of f ( x ) as x approaches c is L . This is written as. - PowerPoint PPT Presentation
Transcript of Section 1.2
Section 1.2
Introduction to Limits
Definition of Limit
If f(x) becomes arbitrarily close to a unique number L as x approaches c from either side, the limit of f(x) as x approaches c is L. This is written as
lim
x cf x L
One way to estimate the limit is numerically. That is , use a table to estimate the limit by looking at values on either side of the value of x in the limit.
Example 1
Use a table and the definition of a limit to estimate numerically the limit:
3lim 5 3x
x x f(x)2.9
2.992.9993.0013.013.1
11.50011.95011.995
12.512.0512.005
12
Example 2
Use a table and the definition of a limit to estimate numerically the limit:
0lim
9 3x
xx
x f(x)-0.01
-0.001-
0.00010.00010.0010.01
5.9985.99985.99998
6.0026.00026.00002
6
Example 3
1, 52, 5
xf x
x
1. Graph this function by hand.
5
Find lim .x
f x
5
lim 1x
f x
1, 52, 5
xf x
x
SOME EXAMPLES OF LIMITS THAT DO NOT EXIST
Example 4
Show that the limit does not exist.
1. Graph the function by hand.HINT: Make a table.
1
1lim 1x
xx
1
1lim 1x
xx
-2 -1 1 2
-2
-1
1
2
Since f(x) as x approaches 1 from the left is -1 and f(x) as x approaches 1 from the right is 1, no limit exists.This is written as
1
1lim does not exist1x
xx
Example 5
Discuss the existence of the limit:
1. Graph the function by hand.HINT: Make a table.
401lim
x x
401lim
x x
As x approaches 0 from the left what happens to f(x)?f(x) increases without bound as x approaches 0 from the left.As x approaches 0 from the right what happens to f(x)?f(x) increases without bound as x approaches 0 from the right.
Since f(x) increases without bound as x approaches 0, you can conclude that the limit does not exist.This is written as
40
1lim does not existx x
Example 6
Discuss the existence of the limit
1. Graph the function on a graphingcalculator.Use -1.5 ≤ x ≤ 1.5 and -1.5 ≤ y ≤
1.5for your window.
01limsin
x x
01limsin
x x
Since f(x) oscillates between -1 and 1 as x approaches 0, the limit does not exist.This is written as
Read Common Types of Behavior Associated with Nonexistence of a Limit on p. 51.HW: pp. 54-56 (2-28 even)
Read pp. 59-61.
01limsin does not exist
x x
