Chapter 13
Limits
Mr. Garis
Chapter 13 Assignments
4/6 M: 13.1 p. 948 1-25 eoo, 29-35 o
T: 13.2 p. 956 1, 3-31 eoo, 33,37,41
W: 13.3 p. 963 1-20, 21-31 o
R: 13.3 p. 964 33-47 o
F: 13.3 p. 964 49-69 o (no graphing), 41,45 (disc)
4/13 M: Trig Limits
T: 13.1,13.2 Quiz / Ch.3 Review
W: Ch.13 Review
R: Chapter 13 TEST
F: 13.4 WS#1 odds
4/20 M: 13.4 WS#2: 1-4 / WS#1: evens
T: 13.4 WS#2: 5-13
W: 14.1 WS#1
R: Review
F: 13.4 Quiz
4/27 M: 14.1 WS#2
T: 14.2 WS#1 Product Rule
W: 14.2 WS#2 Quotient Rule
R: Review
F: Derivative Quiz Early Release Prom
Honors Pre-Calculus
13.1 – Finding Limits Using Tables and Graphs
Learning Targets: Students will be able to find a limit using a table and find a limit using a graph.
If the y-value of a function f gets closer and closer to a particular number N as the x-value gets closer and closer to a
number c from both sides, then that number N is called “the limit of f(x) as x approaches c”
Picture:
Notation:
lim f ( x)  N
x c
N
Spoken as “the limit of f(x) as x approaches c equals N”
c
Finding a limit using a table on the graphing calculator:
On a graph page
Enter the function from the limit in Y1 
Menu, Table (7), Split-screen Table (1) (Or hit control T)
While cursor is on table, Menu, Table (2), Edit Table Settings (5), set Independent to Ask
Enter values for x that approach c from either side (increasing increments by powers of 0.1)
Ex (like 4)
lim
x 3
2.9
2 x

x2  4
2.99
2.999
e x  e x

x 0
2
8. lim
The 3 cases when a limit does not exist (DNE):
1.
The function approaches different
numbers as x approaches c.
2.
The function goes off to   as
x approaches c.
3.
The function oscillates between
numbers as x approaches c.
3
3.001
3.01
3.1
Finding a limit using a graph:
Use the graph to find the indicated limit, if it exists.
12. lim f ( x) 
16. lim f ( x) 
x4
x4
y
y
8
6
3
4
x
2
2
x
4
4
-3
8
(4, -3)
18. If f ( x)  2 x  1, lim f ( x) 
x 1
24. If f ( x)  cos x, lim f ( x) 
x 
Honors Pre-Calculus
13.1 – Finding Limits Using Tables and Graphs Day 2
Learning Target: Students will be able to find a limit using a table and find a limit using a graph.
Find the limit using a table:
 x2  9 
6. lim  2

x 3 x  3 x


14. Find the limit by using the following diagram.
lim f ( x) 
x 3
2
1
1
2
3
 x2 ,
32. Find If f ( x)  
x2
2 x  1, x  2
, find lim f ( x) 
x 2
36.
e x , x  0
f ( x)  
, find lim f ( x) 
x 0
1  x, x  0
Honors Pre-Calculus
13.2 – Finding Limits (Day 1)
Learning Target: Students will be able to find a limit analytically and find the limit of an average rate of
change.
Today we will be looking at an algebraic way to approach limits.
Theorem: If P is a polynomial function, then the lim P ( x)  P (c) for any number c.
x c
Example: lim 5 
x3
lim x 
x 3
Ways to find limits (in this order):
1. Direct substitution
2. Analytically or (Algebraically - do algebra so direct substitution works)
3. Table OR Graph
6. lim  2  5 x  
x 3
20.
 x2  x 
lim  2

x 1 x  1


 3x  4 

2
 x x
16. lim 
x2
 x 3  x 2  3x  3 

2
x

3
x

4


30. lim 
x 1
Now, we can recall what the average rate of change is:
f ( x )  f (c )
xc
34. Find the average rate of change if:
c  2, f ( x)  4  3 x
Find the limit of the average rate of change as x approaches c.
42. Find the rate of change if:
c  1, f ( x) 
1
x2
Find the limit of the average rate of change as x approaches c.
Honors Pre-Calculus
13.2 Other Analytical Methods – Day 2
Learning Target: Students will be able to find a limit analytically and by using the special trig limits.
How do you solve the problem lim
x 0
Ex. lim
x 0
16  x  4
?
x
16  x  4

x
Ex. lim
x 4
x 2

x4
Limits with Trig:
Special Trig Limits MEMORIZE!!!
sin x
1
x 0
x
lim
Ex. lim
x 0
1

x cot x
cos x  1
0
x 0
x
lim
1  cos x
0
x 0
x
OR lim
1  cos 2 x

x 0
x
Ex. lim
Honors Pre-Calculus
13.3 One-Sided Limits and Continuous Functions
Learning Target: Students will be able to find the one-sided limits of a function and determine whether a
function is continuous.
One-Sided Limits:
Sometimes we only consider the limit of f as x approaches c from one side of c only:
The limit of f(x) as x approaches c from the left equals L:
lim f ( x)  L
x c 
The limit of f(x) as x approaches c from the right equals R:
lim f ( x)  R
x c 
“left limit”
“right limit”
Continuous Functions:
A function f is said to be continuous if its graph can be drawn without lifting the pencil from the paper.
Continuity of a function at a point:
A function f is continuous at c if:
1. f(c) is defined; i.e. c is in the domain of f, and f(c) equals a number.
2. lim f ( x)  f (c)
x c
3.
lim f ( x)  f (c)
x c 
In other words, a function f is continuous at a particular point c if lim f ( x)  f (c)
x c
If f is not continuous at c, we say that f is discontinuous at c.
Find the one-sided limit.


24. lim 3x  8 
x 2
28. lim
x 1
2
x3  x

x 1
26.
lim  3cos x  
x  
Ex. Answer the questions using the given graph:
a.
What is the domain of f?
b. What is the range of f?
c.
f ( 3) 
d.
f (2) 
e.
f.
g.
h.
i.
lim f ( x) 
x 3
lim f ( x) 
x2
lim f ( x) 
x 1
lim f ( x) 
x 1
Is f continuous at 0?
Honors Pre-Calculus
13.3 Discontinuity - Day 2
Learning Target: Students will be able to find the one-sided limits of a function and determine whether a
function is continuous.
Test to determine if a function is continuous at a given point c:
1.
2.
3.
f(c) is defined (will have a numeric value when you plug c into the function)
lim f ( x)  f (c)
x c 
lim f ( x)  f (c)
x c 
Determine whether f is continuous at c:
34.
44.
f ( x)  3x 2  6 x  5, c  3
 x2  6x
, x0

f ( x)   x 2  6 x
, c=0
 1,
x0

x2  6 x
, c0
x2  6x
40.
f ( x) 
46.
 x2  2 x
 x2 , x  2

f ( x)  2,
x2, c2
x4

, x2
 x  1
Honors Pre-Calculus
13.3 Discontinuity - Day 3
Learning Target: Students will be able to find the one-sided limits of a function and determine whether a
function is continuous.
Types of Discontinuity
Removable:
expression.
It is a graph that has a hole in it. The hole can be taken away (removed) analytically by simplifying the
c
c
Non-Removable: It is a graph that has asymptotes or gaps in the graph. It can NOT be taken away analytically.
c
c
Find where the function is continuous and/or discontinuous:
50.
f ( x)  4  3 x
56.
f ( x)  4 csc x
58.
f ( x) 
x2  4
x2  9
Where are the following functions continuous? Discuss the continuity (where is it discontinuous and what type).
64. R( x) 
x2  4x
x 2  16
68. R( x) 
x3  x 2  3x  3
x 2  3x  4
Piecewise Functions: Look at all of the pieces and see if there are any holes in each piece. Then look at where one piece
ends and another piece begins.
Continuous Piecewise Function
f (c)  lim f ( x)  lim f ( x)
x c
x c
Removable Discontinuity for Piecewise Function
lim f ( x)  lim f ( x), but  f (c)
x c 
x c
Non-Removable Discontinuity for Piecewise Function
lim f ( x)  lim f ( x)
x c
x c
42. Discuss the continuity:
 x2  6x
, x0

f ( x)   x 2  6 x
2
, x0
