Limit of function

2. Consider the function below and investigate on the values of 𝑓(𝑥) for
the given values of 𝑥.
𝒇 𝒙 =
𝒙𝟑
− 𝒙𝟐
𝒙 − 𝟏
Table A Table B

3. 1.Basedonthetable, whatis impliedonthedomainof 𝑓(𝑥)?
Table A Table B

4. 2. Howwould youdescribethevalues ofthe independent
variable𝑥onTableA?onTableB?
Table A Table B

5. 3. Whatseems tobethevalue of 𝑓(𝑥)as𝑥approaches tothe
valueof 1?
Table A Table B

6. 4.Whatwillhappentothevalue of 𝑓(𝑥)if𝑥 =1?
𝒇 𝟏 =
𝟏𝟑 − 𝟏𝟐
𝟏 − 𝟏
𝒇 𝟏 =
𝟏 − 𝟏
𝟎
𝒇 𝟏 =
𝟎
𝟎
𝒇 𝟏 = 𝒊𝒏𝒅𝒆𝒕𝒆𝒓𝒎𝒊𝒏𝒂𝒕𝒆
Take note:
indeterminate becomes a hole in a graph
𝒇 𝒙 =
𝒙𝟑 − 𝒙𝟐
𝒙 − 𝟏

7. Illustrate a limit of a function (STEM_BC11LC-IIIa-1).

8. Consider a function f(𝒙). Consider a constant 𝒄 which the
variable 𝒙 will approach (𝒄 may or may not be in the domain
of 𝒇).
The limit, to be denoted by 𝑳, is the unique real value that 𝒇(
𝒙) will approach as 𝒙 approaches 𝒄. In symbols, we write this
process as
This is read as “The limit of 𝒇(𝒙) as 𝒙 approaches 𝒄 is 𝑳.”
𝐥𝐢𝐦
𝒙→𝒄
𝒇 𝒙 = 𝑳

9. 1. Table of Values
2. Graph

10. 𝒇 𝒙 =
𝒙𝟑
− 𝒙𝟐
𝒙 − 𝟏
𝐥𝐢𝐦
𝒙→𝒄
𝒇 𝒙 = 𝑳 𝐥𝐢𝐦
𝒙→𝟏
𝒙𝟑
− 𝒙𝟐
𝒙 − 𝟏
= 𝑳
Find the limit of the function, as x approaches 1.
𝐥𝐢𝐦
𝒙→𝟏−
𝒙𝟑 − 𝒙𝟐
𝒙 − 𝟏
= 𝑳 𝐥𝐢𝐦
𝒙→𝟏+
𝒙𝟑 − 𝒙𝟐
𝒙 − 𝟏
= 𝑳
“The limit of f(x) as x approaches
1 from the left”
“The limit of f(x) as x approaches
1 from the right”

11. 𝐥𝐢𝐦
𝒙→𝟏−
𝒙𝟑 − 𝒙𝟐
𝒙 − 𝟏
𝐥𝐢𝐦
𝒙→𝟏+
𝒙𝟑 − 𝒙𝟐
𝒙 − 𝟏
𝐥𝐢𝐦
𝒙→𝟏−
𝒙𝟑 − 𝒙𝟐
𝒙 − 𝟏
= 𝑳
𝐥𝐢𝐦
𝒙→𝟏−
𝒙𝟑 − 𝒙𝟐
𝒙 − 𝟏
= 𝟏
x < 1 f(x)
0.25
0.81
0.9801
0.99980001
0.9999800001
x > 1 f(x)
2.25
1.21
1.0201
1.002001
1.002001
𝐥𝐢𝐦
𝒙→𝟏+
𝒙𝟑 − 𝒙𝟐
𝒙 − 𝟏
= 𝑳
𝐥𝐢𝐦
𝒙→𝟏+
𝒙𝟑
− 𝒙𝟐
𝒙 − 𝟏
= 𝟏
0.5
0.9
0.99
0.999
0.9999
1.5
1.1
1.01
1.001
1.0001

12. 𝐥𝐢𝐦
𝒙→𝟏−
𝒙𝟑 − 𝒙𝟐
𝒙 − 𝟏
= 𝟏 𝐥𝐢𝐦
𝒙→𝟏+
𝒙𝟑 − 𝒙𝟐
𝒙 − 𝟏
= 𝟏
Left-hand Limit: Right-hand
Limit:
Conclusion:
Since the left-hand and right-hand limits are
equal, therefore,
𝐥𝐢𝐦
𝒙→𝟏
𝒙𝟑
− 𝒙𝟐
𝒙 − 𝟏
= 𝟏

13. 𝐥𝐢𝐦
𝒙→𝟏−
𝒙𝟑
− 𝒙𝟐
𝒙 − 𝟏
= 𝟏
𝐥𝐢𝐦
𝒙→𝟏+
𝒙𝟑
− 𝒙𝟐
𝒙 − 𝟏
= 𝟏
Left-hand Limit:
Right-hand
Limit:
𝐥𝐢𝐦
𝒙→𝟏
𝒙𝟑 − 𝒙𝟐
𝒙 − 𝟏
= 𝟏

14. The limit is the unique real
value that 𝒇(𝒙) will approach
as 𝒙 approaches 𝒄.
𝐥𝐢𝐦
𝒙→𝒄
𝒇 𝒙 = 𝑳
𝐥𝐢𝐦
𝒙→𝒄−
𝒇 𝒙 𝐥𝐢𝐦
𝒙→𝒄+
𝒇 𝒙
Left-hand Limit Right-hand
Limit
Judgment:
 If both left-hand and right-
hand limits are EQUAL,
the limit exist, which is L
(value).
 If both left-hand and right-
hand limits are NOT
EQUAL, the limit does not
exist (DNE).

15. 𝒇 𝒙 = 𝟏 + 𝟑𝒙
𝐥𝐢𝐦
𝒙→𝒄
𝒇 𝒙 = 𝑳 𝐥𝐢𝐦
𝒙→𝟐
(𝟏 + 𝟑𝒙) = 𝑳
Find the limit of the function, as x approaches 2.
𝐥𝐢𝐦
𝒙→𝟐−
(𝟏 + 𝟑𝒙) = 𝑳 𝐥𝐢𝐦
𝒙→𝟐+
(𝟏 + 𝟑𝒙) = 𝑳
“The limit of f(x) as x approaches
2 from the left”
“The limit of f(x) as x approaches
2 from the right”

16. 𝐥𝐢𝐦
𝒙→𝟐−
(𝟏 + 𝟑𝒙) 𝐥𝐢𝐦
𝒙→𝟐+
(𝟏 + 𝟑𝒙)
𝐥𝐢𝐦
𝒙→𝟐−
(𝟏 + 𝟑𝒙) = 𝑳
𝐥𝐢𝐦
𝒙→𝟐−
(𝟏 + 𝟑𝒙) = 𝟕
x < 2 f(x)
6.1
6.7
6.85
6.997
6.9997
x > 2 f(x)
8.5
7.3
7.03
7.003
7.0003
𝐥𝐢𝐦
𝒙→𝟐+
(𝟏 + 𝟑𝒙) = 𝑳
𝐥𝐢𝐦
𝒙→𝟐+
(𝟏 + 𝟑𝒙) = 𝟕
1.7
1.9
1.95
1.999
1.9999
2.5
2.1
2.01
2.001
2.0001
Table A Table B

17. 𝐥𝐢𝐦
𝒙→𝟐−
(𝒙 + 𝟑𝒙) = 𝟕 𝐥𝐢𝐦
𝒙→𝟏+
(𝟏 + 𝟑𝒙) = 𝟕
Left-hand Limit: Right-hand
Limit:
Conclusion:
Since the left-hand and right-hand limits are
equal, therefore,
𝐥𝐢𝐦
𝒙→𝟐
(𝟏 + 𝟑𝒙) = 𝟕

18. 𝐥𝐢𝐦
𝒙→𝟐−
(𝟏 + 𝟑𝒙) = 𝟕
𝐥𝐢𝐦
𝒙→𝟐+
(𝟏 + 𝟑𝒙) = 𝟕
Left-hand Limit:
Right-hand
Limit:
𝐥𝐢𝐦
𝒙→𝟏
(𝟏 + 𝟑𝒙) = 𝟕

19. Using the graph, find the limit of the function as x
approaches from the following values.
1. 𝑥 = −1
2. 𝑥 = 3
3. 𝑥 = −3.75
4. 𝑥 = 1
5. 𝑥 = 5

20. 1. Form a circle with your group mates.
2. Prepare your pens and calculators.
3. As you received the material from the
teacher, provide the necessary answer.
4. If you are already done answering, give
the material to the teacher.
5. Prepare your group for a presentation
later.

21. 𝒈 𝒙 =
𝒙𝟐
− 𝟐𝒙
𝒙 − 𝟐
𝒉 𝒙 = 𝒙 + 𝟏
𝒓 𝒙 = 𝒙𝟐 − 𝟒
𝒇 𝒙 =
𝟏
𝒙

22. WhatdowemeanbytheLimitofaFunction?
HowcanwefindtheLimit ofaFunction?
Isitimportanttofindfirsttheleft-hand and
right-handlimits beforestatingtheanswerof
the ?
lim
𝑥→𝑐
𝑓(𝑥)

23. Directions: Get 1 whole yellow paper per group. Copy
and answer only. Do not include the graph in your
paper.
A. Construct two tables of values to find the .
B. Consider the function 𝑓(𝑥) whose graph is given below.
1.
2.
3.
4.
5.
lim
𝑥→1
(𝑥3
+2𝑥)
lim
𝑥→−3
𝑓 𝑥 =
lim
𝑥→1
𝑓 𝑥 =
lim
𝑥→3
𝑓 𝑥 =
lim
𝑥→2
𝑓 𝑥 =
lim
𝑥→4
𝑓 𝑥 =

24. Directions: Get 1 whole yellow paper per group. Copy
and answer only. Do not include the graph in your
paper.
A. Construct two tables of values to
find the
B. Consider the function 𝑓(𝑥) whose
graph is given below.
1.
2.
3.
4.
5.
lim
𝑥→1
(𝑥3+2𝑥)
lim
𝑥→−3
𝑓 𝑥 =
lim
𝑥→1
𝑓 𝑥 =
lim
𝑥→2
𝑓 𝑥 =
lim
𝑥→6
𝑓 𝑥 =
lim
𝑥→4
𝑓 𝑥 =

25. Arceo, Carlene Perpetua P., Richard S. Lemence, Oreste Jr. M. Ortega, and Louie John D.
Vallejo. (2016). Teaching Guide For Senior High School Basic Calculus. Diliman,
Quezon City: Comission on Higher Education

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