Name: __________________________________ Period _______ Sec1.2 Definition of Limit
Sec1.3 Evaluating limits analytically
Assignment: Sec1.2 Page 56-57: # 37, 39, 41, 49, 51
Sec1.3 Page 67: # 1,5,9,13,17,21,25,29, 33, 37, 44 (Every other odd ….eoo)
WARMUP: Before getting started with calculus we need to review
how we write distances.
The distance a number is on the x-axis away from another number,
c, is written using absolute values: | x  c |
For example: | x  2 | = the distance an x-value is away from 2
Select several values of x to verify that this notation works.
Similarly, | y  L | =distance a y-value is away from L
For example: | y  5 | =the distance a y-value is away from 5
Select several values of y to verify that this notation works.
Objective: In this lesson you learn the formal definition of limit and
how to evaluate a limit using Limit properties
What you should learn
Formal definition of limit
Informal definition of a limit: If
f ( x ) becomes arbitrarily close to a single number L as x approaches c
from either side, then the limit of f ( x) as x approaches c is L, and is written as lim f ( x)  L
xc
Formal definition of limit: The epsilon-delta definition is written using the greek letters epsilon) to
represent distances on the y-axis and  (delta) to represent distances on the x-axis.
Look at the first part of the informal definition: “ f ( x) becomes arbitrarily close to a single number L”,
This means that the y-value on the function is in the interval: L - e < y-value on the function < L + e
We can write this using absolute value notation as
| f ( x)  L | 
Similarly, the second part of the informal definition, “x approaches c”, means that the x-value is either
on the left side of c; in the interval c- < x-value < c, or
on the right side of c, in the interval c < x-value < c + 
We can write this using absolute value notation as 0 | x  c | 
This distance is greater than zero (so x can not equal c) and
is less than delta.
This creates a “box” around the limiting value on the function.
 –  Definition of a limit:
lim
xc
f ( x)  L
means that for any  > 0, there exists a  > 0, such that
If, 0 | x  c |  then | f ( x )  L | 
This means that if the x-value is delta close to c, but x  c then the y-value will be epsilon close to the limit.
Using the  –  Definition of Limit:
#35 The graph of
f ( x)  x  1
is shown in the figure. Find  such that if
If, 0 | x  2 |  then | f ( x)  3 | 0.4
Solution:
Compare to the definition:
If,
0 | x  c | 
then |
f ( x)  L | 
c = _______,  = ________, L = ________ and  = _______
Use your graphing calculator: y1  x  1
y2  L  
y3  L  
Find the points of intersection
Find distance on both the right and left side of c
Delta is the smaller of these two distances. Explain why
#36 The graph of f ( x)  1 is shown in the figure. Find  such that if
x 1
Solution:
Compare to the definition:
If,
0 | x  2 | 
then
If,
0 | x  c | 
then |
| f ( x)  1| 0.01
f ( x)  L | 
c = _______,  = ________, L = ________ and  = _______
Use your graphing calculator: y1  1
x 1
y2  L  
y3  L  
Find the points of intersection
Find distance on both the right and left side of c
Delta is the smaller of these two distances. Explain why
#42 Find the limit L = lim
x2  4
x5
Then, find  such that
| f ( x)  L | 0.01
whenever
0 | x  c | 
c = _______,  = ________, L = ________ and  = _______
Use your graphing calculator: y1  x2  4
y2  L  
y3  L  
Find the points of intersection
Find distance on both the right and left side of c
Delta is the smaller of these two distances. Explain why
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Evaluating Limits Using Limit Properties (see properties …Theorems 1.1 to 1.6 on pages 59 to 61).
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Luckily, the limit properties allow us to evaluate most limits by directly substituting in the value of c.
What this means is that when you substitute the value of c, you get a real number. That real number is
the limit. The limit can also be verified with a graph.
What you should learn?
Examples: 3, 7, 11, 15, 19, 23, 27, 31, 35, 39, 43
How to evaluate a limit by
substituting the value of c
into the function.