1.1 Limits packet

```Calculus
thoughts here!
1.1 Limits Graphically
Name:_________________________
Notes
What is a limit?
A limit is the __________ a function ______________ from both the left
and the right side of a given ___________.
Example 1

y

y










     

3

     

3
lim f x  
lim f x  
x3
x3
y





y



x
     

3
lim f x  
x3

x
x
     

y
x

     

3

3
lim f x  
lim f x  
x3
x3
Limit: (geeky math definition for Mr. Kelly)
Given a function , the limit of
as approaches is a real number if
can
be made arbitrarily close to by taking sufficiently close to (but not equal to ). If
.
the limit exists and is a real number, then the common notation is
→
What is a one‐sided limit?
A one‐sided limit is the ___________ a function approaches as you
approach a given ___________ from either the ______ or ______ side.
Example 2
y


“The limit of as approaches 3
from the left side is 1.”
lim
“The limit of as approaches 3
from the right side is 2.”
lim
1
→






→
2
thoughts here!
1.1 Limits Graphically
Example 3
a. lim f  x  
b. lim f  x  
x  2 
x  2
Notes
c. lim f  x  
y

x2

d. lim f x  
e. lim f  x  
f. lim f  x  
g. lim f  x  
h. lim f  x  
i.
x 1
x 0
x  1
x  3


x 3

x
2














j.
1



When does a limit not exist?
1.
2.
3.
Example 4
Sketch a graph of a function that
satisfies all of the following conditions.
a.
3
1
b. lim g ( x)  4
y






x3
c.
lim g ( x)
x
1
      

x 2 
d.
e.
is increasing on 2
lim g ( x) lim g ( x)
x 2

x2





d. lim f  x   2
e. lim f  x   does not exist
x2

c. lim f  x   1
y

x1

x 1

f. lim f x   lim f  x 
x0



b. lim f  x   2
x0


a. lim f  x   1
x1


3
Example 5
Write T (true) or F (false) under each statement.
Use the graph on the right.
x 1

g. lim f x   does not exist
x 2




1.1 Limits Graphically
Calculus
Name: _____________________________
Practice
For 1‐5, give the value of each statement. If the value does not exist, write “does not exist” or “undefined.”
1.

a.
lim f ( x) 
b.
1
c.
x 1
d.
lim f ( x) 
lim f ( x) 

x 0
1
e.
f.
y


2
x
x 2 









g.
lim f ( x) 
h.
x 1
lim f ( x) 
i.
x 1
lim f ( x) 

x 2


2.

a.
lim f ( x) 
b.
1
c.
x  3
lim f ( x) 
y

x 1


d.
lim f ( x) 
x 2
e.
3
f.

lim f ( x) 
x 2

x

















g.
lim f ( x) 
2
h.
i.
4


x  2


3.

a.
lim f ( x) 
b.
3
c.

lim f ( x) 

x 0
x 3
y


d.
lim f ( x) 
e.
0
f.
lim f ( x) 
x 3
x 3

x






g.
lim f ( x) 
h.
1
1.6
i.


x 0


4.

a.
lim f ( x) 
b.
2
c.
x 1
lim f ( x) 
y

x2


d.
lim f ( x) 
e.
4
f.
g.
lim f ( x) 

x 1
lim f ( x) 


x  1

x 1
h.
1
i.
lim f ( x) 
x 4
x














y
5.
a.
lim f ( x) 
1
b.
c.
x  3

lim f ( x) 

x  3

d.
lim f ( x) 
3
e.
f.
x  1
lim f ( x) 
x 3



x
g.
3
h.
lim f ( x) 
i.

4









x 0


6. Sketch a graph of a function
following conditions.
a.
2
5
b.
that satisfies all of the
y



lim f ( x)  1


x 2

x
c.
lim f ( x)
x 4
3
      

























d.
is increasing on

2


e.
lim f ( x) lim f ( x)
x 4
x 4 
7. Sketch a graph of a function
following conditions.
a.
1
3
that satisfies all of the
y


b.
lim g ( x)  2


x1


c.
lim g ( x)
x
5
      

x3

is increasing only on 5
d.
3 and
1



e.

lim g ( x) lim g ( x)
x3
x3
8. Sketch a graph of a function that satisfies all of the
following conditions.
a. limh( x)
2
1
x3
y





c. lim
→
d.

3 is undefined.
b.
x
      


lim

→
is constant on 2
everywhere else.

3 and decreasing



Test Prep
1.1 Limits Graphically
1. The graph of the function f is shown. Which of the following statements about f is true?
(A) lim
lim
(B) lim
4
(C) lim
4
(D) lim
1
→
→
→
(E) lim
2. The figure below shows the graph of a function
statements are true?
lim
exists.
II.
lim
exists.
III.
lim
→
→
→
→
does not exist.
→
I.
→
with domain 0
4. Which of the following
exists.
(A) I only
(B) II only
(C) I and II only
(D) I and III only
(E) I, II, and III
represents the greatest integer that is less than or equal to x, then lim
3. If
→
(A)
2
(B)
4. Consider the function
1
(C) 0
(D) 2
(E) the limit does not exist
shown below. Which of the following statements is true?
(A) lim
3
→
(B)
1
(C)
1
is continuous for all x.
(D) lim
→
1
(E) None of the above
```