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Calculus AB
Chapter 1 - Limits
Mrs. Boddy
8/31/2015
AP Calculus AB
Practice – Chapter 1
Date
Day
Section
8/31
M
1
1.2/1.3
9/1
T
2
1.2/1.3
9/2
W
3
1.5


Determine horizontal asymptotes
Find limits at horizontal
asympototes
Assignment
p. 55 - 58 9, 15-23 odd, 26, 29, 64,
71-76
p. 67 11, 17, 21, 25, 29, 31, 35
p. 67 37, 40, 49-61 odd, 65,
71-75 odd, 85, 116-122
p. 88 & 89 39, 43, 45, 47, 49, 61, 63
**Online Practice Quiz – 1.1-1.3**
9/3
Th
4
9/4
F
5
9/8
T
6
1.4
9/9
W
7
1.4
9/10
Th
8
Review
AP 1-1 1,3,5 (in book after p. 94)
9/11
F
9/14
M
9
Review
WS – Review
Review
Review
9/15
T
3.5
Learning Target
What is a limit?
 Find using a graph
 Find using a table
 Find using direct substitution
 Algebraic techniques for finding
limits
o factor
o divide
o rationalize
 Trigonometric Limits
 Determine vertical asymptotes
 Find limits at asymptotes
p. 205 1-6, 15, 19, 29, 31
**Summative Quiz – 1.2, 1.3, 1.5**

Test for continuity using left and
right limits

Test for continuity using left and
right limits
Apply the Intermediate Value
Theorem

p. 78 1-13odd, 19, 21,27,35-45odd,
51, 63, 64
Worksheet 1.4
10
11
CHAPTER 1 TEST
**Quiz – 1.4**
Section 1.2-1.3
Finding Limits
Learning Standard:
377.01 Students will be able to evaluate limits and determine the continuity of a function.
1A
Evaluate limits by many methods
1B
Evaluate one-sided limits of functions
What is 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, written as lim f ( x)  L .
x c
Find a limit using a table
• go to graph page, enter function
• go to menu, table (7), split screen
• go to menu, table (2), Table Set (5)
• independent – ask
x3
lim 2
• Try:
x 3 x  7 x  12
(or Control T)
x
y
Find a limit using a graph
lim f  x 
x 7
lim f  x 
x 0
lim
f  x
x 5
lim sin x
x
3
2
 4, x  0

lim 2 x, 0  x  2
x 2
  x2 , x  2

Where do limits NOT exist (DNE)



Some basic limits that you need to know
• Limit of a constant
lim 5 
x3
•
Limit of the identity function
lim x 
•
Limit of a power function
lim x 2 
x 3
x 3
Find a limit using direct substitution
• Substitute the limit value directly into the expression to calculate the result.
2x  2
lim
• Try:
lim sec x
x2 x  3
x 
One-Sided Limits – Graphically
One-Sided Limits – Analytically
lim f  x  
x 1
x  1
 x  1,
f  x   2
 x  2 x, x  1
lim f  x 
lim f  x  
x 1
lim f  x  
x 1
x 1
lim f  x 
x 1
lim f  x  
x 6
lim f  x  
x  6
lim f  x  
x 6
lim f  x 
x 1
Section 1.3
Finding Limits – Day 2
Find the following limits algebraically.
lim
x 2  16
x4 x  4
lim
3x 2  4x  1
x1
x2  1
x3  1
x1 x  1
lim
lim
x 0
x 1 1
x
Special Trigonometric Limits (MEMORIZE ME!!!)
sin x
1
x 0
x
x
lim
1
x  0 sin x
lim
Find these trigonometric limits
cos x tan x
lim
x 0
x
1  cos x
0
x 0
x
cos x  1
lim
0
x 0
x
lim
x2
lim
x0 1  cos 2 x
sin 3x
x 0
x
lim
lim
x 0
sin 5 x
sin 4 x
Expansion of Trigonometric Limits (Shortcuts)
sin ax

x 0
x
lim
sin ax

x 0 sin bx
lim
Section 1.5
Infinite Limits
Learning Targets
 Be able to determine infinite limits
 Be able to apply the properties of infinite limits
 Continue to determine limits of functions (use left and right limits)
Definition of Infinite Limits
 A limit in which f (x) increases or decreases without bound as x approaches a number c.
Note:
 This does not mean that the limit equals ∞ - you cannot actually get to infinity. What it tells you is
how the graph behaves at that x value.
 We are going to see infinite limits when we have vertical asymptotes.
Find the vertical asymptotes.
f ( x) 
2x  3
x 2  25
Find the limits.
x
lim
x2 x  1
Graph:
lim 
x   2
lim 
x   2
lim 
x  2
f ( x) 
lim
x 3
1
x2
x2  9
x3
lim
x 4
f ( x) 
x4
x4
1
( x  2) 2
Properties of Infinite Limits
Let c and L be real numbers and let f and g be functions such that
and lim f (x)  
xc
lim g(x)  L
xc
lim  f ( x)  g ( x)   
x c
lim  f ( x) g ( x)   , L  0
x c
lim  f ( x) g ( x)   , L  0
x c
f ( x)

g ( x)
g ( x)
lim
0
x c f ( x )
lim
x c
Examples (What if no calculator?):
1. Find the limit:
3. Find the limit:
x3
lim
x2 x  2
lim
x1
x2  x
x  1x  1
2
2. Find the limit:
4. Find the limit:
x2
lim 2
x4 x  16
lim
x2
3
x  2
2
AP Calculus
Notes – 3.5
Learning Targets:



Determine (finite) limits at infinity
Determine the horizontal asymptotes, if any, of the graph of a function.
Determine infinite limits at infinity
We have been looking at limits.
1
Find : lim 
x0 x
lim
x0
1

x
These limits approached infinity. Today we are going to look at the x value as it approaches infinity.
lim f ( x)  L , then y is approaching a finite value as x is approaching infinity.
x 
Graphically, this means f has a HORIZONTAL ASYMPTOTE at y = L.
Examples:
1.
2 

lim  5  2 
x 
x 

2.
 5 x3  2 x 
lim  3

x 
 3x  4 
3.
 3x  4 
lim  2

x  x  2


4.
lim
x 
x
x 1
2
Remember last year you found the horizontal
asymptotes by looking at:
deg numerator  degdenominator then HA is y  0
a
deg numerator  degdenominator then HA is y 
b

AP Calculus AB
Notes – Sec. 1.4
Learning Targets
 Be able to evaluate continuity of a function using left and right hand limits
Definition of continuity
A function f(x) is continuous at x=c if all of the following conditions exist:
1. The function has a value at x=c, i.e., f(c) exists.
2. The limit exists at c.
3. The limit at c is f(c).
Example 1
Is the function continuous at x=2?
 x  1, x  2
f x   
2x 1, x  2

Does f(2) exist?

Does the limit exist at x=2?

Does the limit equal f (2)?
Example 3
Is the function continuous at x=2?
f  x  x

Does f (2) exist?

Does the limit exist at x=2?

Does the limit equal f(2)?
Example 2
Is the function continuous at x=2?
 x  1,
f x   
2x  1,
x 2
Does f(2) exist?
 Does the limit exist at x=2?
Does the limit equal f (2)?
Example 4
Discuss the continuity of f  x   25  x 2
over the closed interval  5,5
Types of Discontinuities
 Removable discontinuity – hole

x 2
Nonremovable discontinuity - vertical asymptote or break
Examples of Discontinuities
5.
2 x 2  7 x  15
Where does the function f  x   2
have:
x  x  20
a) a nonremovable discontinuity; and
b) a removable discontinuity
6.
 ax, x  1
For what value of a is the function f  x   
continuous at x =-1?
 x, x  1
7.
ax  3, x  3
For what value of a is the function f  x    2
continuous at x = 3?
 x  x, x  3
8.
 x 2  ax, x  2
For what value of a is the function f  x   
continuous at x = -2?
2 x  2, x  2
AP Calculus AB
Notes – Sec. 1.4 cont.
Learning Targets
 Be able to apply the Intermediate Value Theorem to closed interval functions.
Intermediate Value Theorem
If f is continuous on the closed interval [a,b] and k is any number between f(a) and f(b), then there is at
least one number c in [a,b] such that f(c)=k.
a
b
1. Verify that the IVT applies to the indicated interval and find the value of c guaranteed by the
theorem.
f  x   x2 1, 0,3 , f(c) = 3 (Hint: Verify that the IVT applies, then solve f(x) = 3)
2. Explain why the function f  x   x3  5x  3 has a zero in the interval 0,1 .