Calculus
Write your questions and
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
x3
x3
y
y
x
3
lim f x
x3
x
x
y
x
3
3
lim f x
lim f x
x3
x3
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
Write your questions and
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
x2
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
x3
c. lim g ( x)
x
1
x 2
d.
e.
is increasing on 2
lim g ( x) lim g ( x)
x 2
x2
d. lim f x 2
e. lim f x does not exist
x2
c. lim f x 1
y
x1
x 1
f. lim f x lim f x
x0
b. lim f x 2
x0
a. lim f x 1
x1
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.
c. lim f ( x)
1
x 1
d. lim f ( x)
x 0
1
e.
f.
y
2
x
x 2
g. lim f ( x)
x 1
h. lim f ( x)
x 1
i. lim f ( x)
x 2
2.
a. lim f ( x)
b.
c. lim f ( x)
1
x 3
y
x 1
d. lim f ( x)
x 2
e.
f. lim f ( x)
3
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
x 1
c. lim f ( x )
y
x2
d. lim f ( x)
e.
4
g. lim f ( x)
x 1
f. lim f ( x)
x 1
x 1
h.
1
i. lim f ( x)
x 4
x
y
5.
a. lim f ( x )
c. lim f ( x)
1
b.
x 3
x 3
d. lim f ( x)
f. lim f ( x)
3
e.
x 1
x 3
x
g.
h. lim f ( x)
3
i.
4
x 0
6. Sketch a graph of a function
following conditions.
a.
2
5
that satisfies all of the
y
b. 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
x1
c. lim g ( x)
x
5
x3
is increasing only on 5
d.
3 and
1
e.
lim g ( x) lim g ( x)
x3
x3
8. Sketch a graph of a function that satisfies all of the
following conditions.
a. limh( x)
2
1
x3
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
exists.
→
→
→
(A) I only
(B) II only
→
does not exist.
→
I.
→
with domain 0
(C) I and II only
4. Which of the following
(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
0
