Calculus
1.5 Algebraic Properties of Limits and Piecewise Functions
Notes
Write your questions
and thoughts here!
𝑥
lim 𝑓 𝑥
→
2
2𝑥
𝑓 𝑥
→
Example 1:
lim 𝑓 𝑥
𝑥
2 lim 𝑓 𝑥
→
lim 𝑓 𝑥
→
4
lim 𝑔 𝑥
6
→
The table above gives selected limits of the functions 𝑓 and 𝑔. What is lim 𝑓
𝑥
→
Example 2:
The graph of the function 𝑓 is shown on the
right. What is lim 𝑓 𝑓 𝑥 ?
?
y

→









x






Graph of 𝑓
Example 3:
𝑓 5
1
𝑔 5
2
ℎ 5
3
lim 𝑓 𝑥
→
lim 𝑔 𝑥
→
lim ℎ 𝑥
→
6
1
5
The table above gives selected values and limits of the functions 𝑓, 𝑔, and ℎ.
What is lim ℎ 𝑥 𝑓 𝑥
2𝑔 𝑥
ℎ 5 ?
→
Example 4: Piecewise Functions
Piecewise defined functions and limits
√11
𝑓 𝑥
a.
lim 𝑓 𝑥
→
c. lim 𝑓 𝑥
→
𝑥, 𝑥
𝑥 3
,
5 𝑥
b.
𝑥
5
lim 𝑓 𝑥
→
𝑔 𝑥
5
a. lim 𝑔 𝑥
→
c. lim 𝑔 𝑥
→
⎧ 10 𝑥 ,
⎪
26 5𝑥
⎨
7
⎪
ln
𝑥 ,
⎩
𝑥
1
1
𝑥
𝑥
𝑒
𝑒
b. lim 𝑔 𝑥
→

1.5 Algebraic Properties of Limits
Practice
Calculus
Use the table for each problem to find the given limits.
1.
4
lim 𝑓 𝑥
2
lim 𝑓 𝑥
→
→
a. lim 2𝑓 𝑥
𝑔
→
𝑥
lim 𝑔 𝑥
1
→
b.
lim 𝑔 𝑥
5
→
lim
→
2.
lim 𝑓 𝑥
a. lim 𝑓 2𝑥
lim 𝑓 𝑥
1
→
→
6
𝑓
→
lim 𝑓 𝑥
lim 𝑓 𝑥
2
→
3
→
b. lim
→
Use the graph for each problem to find the given limits.
4.
3.

y


y






x









x















Graph of 𝑓
Graph of 𝑓
a. lim 𝑓 𝑓 𝑥
a.
b. lim 𝑓 𝑓 𝑥
b. lim 𝑓 𝑓 𝑥
→
→
lim 𝑓 𝑓 𝑥
→
→
Use the table for each problem to find the given limits.
5.
𝑓 1
a. lim
→
𝑓 𝑥
lim 𝑓 𝑥
4
1
→
𝑔 1
2
lim 𝑔 𝑥
3
ℎ 1
3
lim ℎ 𝑥
6
ℎ 𝑥
𝑔 1
→
→
b. 𝑓 1
lim
→
𝑔 𝑥



6.
𝑓
2
7
𝑔
2
1
ℎ
2
lim ℎ 𝑥 2𝑓 𝑥
a.
𝜃
h. 𝑔
g. ℎ
h. lim ℎ 𝑥
4
2𝜋
2𝜋
c. lim 𝑤 𝜃
d.
e. lim 𝑤 𝜃
f.
g. lim 𝑤 𝜃
h. 𝑤 2𝜋
→
b. lim ℎ 𝑥
f. lim ℎ 𝑥
b. 𝑤 𝜋
→
lim ℎ 𝑥
→
→
lim ℎ 𝑥
→
→
a. lim 𝑤 𝜃
→
𝑥
5
5 𝑥 3
𝑥 3
e. lim ℎ 𝑥
→
→
→
5
→
𝜋
𝜃
ℎ 𝑥
→
|𝑥|,
20 𝑥 ,
4𝑥 1,
8. ℎ 𝑥
→
𝜃
2 lim 𝑔 𝑥
f. lim 𝑔 𝑥
g. lim 𝑔 𝑥
→
b. 𝑓
d.
e. lim 𝑔 𝑥
𝜋
3
→
c. ℎ 3
d.
sin 𝜃 ,
cos 𝜃 ,
tan 𝜃 ,
lim ℎ 𝑥
lim 𝑔 𝑥
c. 𝑔 2
9. 𝑤 𝜃
1
a.
b.
→
lim 𝑔 𝑥
lim 𝑔 𝑥
a. lim 𝑔 𝑥
→
2
→
2
Use the piecewise functions to find the given values.
𝑥
4
√5 𝑥,
7. 𝑔 𝑥
5,
4 𝑥 2
𝑥
𝑥 3,
𝑥 2
→
→
4
ℎ
→
lim 𝑓 𝑥
,
10. 𝑓 𝑥
2 ,
𝑥
a. lim 𝑓 𝑥
→
lim 𝑓 𝑥
𝑥
2
4,
2
𝑥
𝑥
0
0
b.
lim 𝑓 𝑥
→
d. lim 𝑓 𝑥
lim 𝑤 𝜃
c.
lim 𝑤 𝜃
e. lim 𝑓 𝑥
f. lim 𝑓 𝑥
g. 𝑓
h. 𝑓 0
→
→
→
→
2
→
→
Test Prep
1.5 Algebraic Properties of Limits
11. If 𝑓 is a continuous function such that 𝑓 3
7, which of the following statements must be true?
(A) lim 𝑓 3𝑥
9
(B) lim 𝑓 2𝑥
(D) lim 𝑓 𝑥
49
(E) lim 𝑓 𝑥
→
→
12. If 𝑓 𝑥
(A) ln 9
13. If 𝑓 𝑥
(A) ln 2
ln 3𝑥 ,
𝑥 ln 3 ,
0
3
→
𝑥
𝑥
→
(B) ln 8
(C) lim
→
7
49
3
, then lim 𝑓 𝑥 is
4
→
(B) ln 27
ln 𝑥
for 0
𝑥 ln 2 for 2
14
(C) 3 ln 3
𝑥
𝑥
(D) 3
ln 3
(E) nonexistent
2
then lim 𝑓 𝑥 is
4,
→
(C) ln 16
(D) 4
(E) nonexistent