Limits and their Properties
Lesson 3.2
Let’s Start With a Question
Given the following
function…
f ( x) 
x  3x  2
x2
2
…evaluate f (2) .
Easy Peasy!
f ( x) 
( x  2)( x  1)
( x  2)
f ( x)  ( x  1)
f (2)  (2)  1  1
f (2) = 0 ÷ 0
…which equals…
???
Let’s Explore
x 2  3x  2
f ( x) 
x2
As x approaches 2 from the left…
And as x approaches 2 from the right…
x
1.9
1.99 1.999
2
2.001 2.01
2.1
f (x)
.9
.99
?
1.001 1.01
1.1
f (x) approaches 1
.999
f (x) approaches 1
So, f (2) = 1, right?
But…
Let’s Define!
Limit – Given f (x), a limit, L, exists if f (x) becomes
arbitrarily close to a single number L as x
approaches c from either side
lim f ( x)  L
x c
“The limit of f (x) as x approaches c is L”
x 2  3x  2
f ( x) 
x2
So, f (2) is not defined, but… lim f ( x)  1
x2
A Quick Word…
The previously aforementioned definition of a limit is not well defined nor precise
enough for rigorous use within the mathematical community. The formal definition,
otherwise known as the “- Definition of a Limit,” is complex and difficult to grasp.
The study of this definition is not required for the AP exam, so its instruction is
therefore omitted from the curriculum. By no means does this omission diminish the
definition’s importance, usefulness, or mathematical perfection.
Anybody interested in understanding more, especially those considering to major in
mathematics in college, is welcome to inquire during non-classtime hours.
Thank you. And now back to your regularly-scheduled lesson.
Let’s Practice
x2 1
1. Find lim
. Use a table to assist you.
x 1 x  1
As x approaches 1 from the left…
x
0.9
0.99 0.999
1
1.001 1.01
1.1
f (x)
1.9
1.99 1.999
2
2.001 2.01
2.1
f (x) approaches 2
2.
As x approaches 1 from the right…
 x  1, x  2
f ( x), f ( x)  
Find lim
x 0
0, x  2
using the graph to assist you.
lim f ( x)  1
x 0
f (x) approaches 2
Let’s Practice Some More
1.
| x|
Find lim
x 0 x
using the graph to assist you.
As x approaches 0 from the left…
f (x) approaches -1
As x approaches 0 from the right…
f (x) approaches 1
Limit does not exist!
Classwork
Complete the table, and use the result to estimate the limit.
8x  2 x 2
lim f ( x) 
x 4
x4
x
f (x)
3.9
3.99 3.999
4
4.001 4.01
?
Homework
Pg. 215 (2, 4, 6, 7-12, 41, 43, 44)
4.1