A grade 12 STEM Capstone presentation

1. INTRODUCTION
OBJECTIVES
LESSON AND COVERAGE
Lesson : LIMITS
In this module, the following lessons will be discussed:
1b
You are now in Grade 12, in this level and the higher level of education, you
might ask the question: What are the significance of Limits in fields of
Mathematics?
At the end of this lesson, you were able to :
 To learn the different concepts of limits and the different
limits rules;
 To understand the different limits theorem; and
 To analys the difference the infinite limits and the limits of
infinity.

2. Lesson 1. Limits
1.A. Methods of finding the Limits of a Function.
1.B. Limit Theorems
1.C. Infinite Limit and Limit of Infinity
In this lesson, you will learn to:
Lesson 1.A Apply the methods of finding Limits of
a Function using the Table Of Values
and Using the Graphs.
Lesson 1.B Identify what are the Theorem used to
Limits and how it solve using its own
Property.
Lesson 1.C Identify what are the Theorem and
how it is solve.
PRE- ASSESSMENT
Let us find out first what you already know about the content of this
module. Try to answer all items. Take note of the questions that you where able
answer correctly and revisit them as you go through this module for self-
correction.
INSTRUCTION:
A. Write True if the answer is Correct and False if it is Not.
1. lim
𝑥→8
𝑥 = 8𝑥 _______________________
2. lim
𝑥→100
𝑥 = 100 _______________________
3. lim
𝑥→−4
( 𝑥 + 12) = 8 _________________
4. !=lim
𝑥→3
5 = 3 _______________________
5. lim
𝑥→6
4𝑥 = 24 _______________________
6. lim
𝑥→−2
√𝑥4 + 4𝑥2 − 7𝑥 + 5
3
= √513
_________________
7. lim
𝑥→16
√ 𝑥4
= 5 _______________________
8. lim
𝑥→−1
𝑥
𝑥−1
=
2
6
_________________
9. lim
𝑥→2
(𝑥2
− 2𝑥 + 3)5
= 243 _________________
10. lim
𝑥→−1
(𝑥2
− 4) = −36 _______________________
B. Identify the following. IL if it is Infinite Limit an LI if it is Limits of Infinity.
_______1. Lim
𝑥→0+
1
𝑥 𝑛 = + ∞
_______2. lim
𝑥→−∞
−2𝑥7
= +∞
2

3. _______3. lim
𝑥→−∞
−21𝑥3
= +∞
_______4. lim
𝑥→0−
1
𝑥7 = −∞
_______5. lim
𝑥→−∞
21𝑥3
= −∞
At the end of this lesson the students well be able to:
1. Discuss the basics of Limits; and
2. Apply the Limits into your real-life situation.
Divide the class into five groups, then follow the mechanics of the activity.
Mechanics:
1. Group members will thinks what are the significance of Limits in our
real-life.
2. The assign leader will pick the best example of limits and the rest of
the members will draw the chosen example.
3. Every group will present their work infront of the class.
Criteria:
Content_____________________________________5points
Appropriateness_____________________________5points
Examples____________________________________5points
Uniqueness __________________________________5points
INVESTIGATION
3b
Lesson 1: LIMITS
OBJECTIVES:
Opening
Activity

4. Limit (Backbone of Calculus)
Consider a function of a single variable x. considere a constant c which the
variable x will approach c. C may or may not be the domain of f(x). The limit to be
denoted by L is the unique real value that f(x) will approach as x approaches c. In
symbol, we write this process at;
𝐥𝐢𝐦
𝒙→𝒄
𝒇( 𝒙) = 𝑳
USING THE TABLE OF VALUE
LOOKING AT A TABLE OF VALUES
To illustrate, let us consider
lim
𝑋→2
(1 + 3𝑥)
(1 + 3x). Here, f(x)=1+3x and the constant c, which x will approach, is 2. To
evaluate the given limit, we will make use of a table to help us keep track of the
effect that the approach of x toward 2 will have on f(x). Of course, on the number
line, x may approach 2 in two ways: through values on its left and through values
on its right. We first consider approaching 2 from its left or through values less
than 2. Remember that the values to be chosen should be close to 2.
Lesson 1.A. Methods of finding the
Limits of a Function.
OBJECTIVES:Apply the methods of finding Limits
of a Function
 Using the table of values
 Using the graphs
4b

5. Solution:
2
x 1.7 1.8 1.9 1.99 2.01 2.05 2.1 2.2
y 6.1 6.4 6.7 6.97 7.03 7.15 7.3 7.6
L = 7
If x = 1.7 if x = 1.8 if x = 1.9 if x = 1.99
y= 1 + 3x y= 1 + 3x y= 1 + 3x y= 1 + 3x
= 1 + 3 (1.7) = 1 + 3 (1.8) = 1 + 3 (1.9) = 1 + 3 (1.99)
= 1 + 5.1 = 1 + 5.4 = 1 + 5.7 = 1 + 5.97
= 6.1 = 6.4 = 6.7 = 6.97
If x = 2.01 if x = 2.05 if x = 2.1 if x = 2.2
y= 1 + 3x y= 1 + 3x y= 1 + 3x y= 1 + 3x
= 1 + 3 (2.01) = 1 + 3 (2.05) = 1 + 3 (2.1) = 1 + 3 (2.2)
= 1 + 6.03 = 1 + 6.15 = 1 + 6.3 = 1 + 6.6
= 7.03 = 7.15 = 7.3 = 7.6
USING THE GRAPH
Using the table of value answered on the last page, put it on the graphs to find
the limit.
lim
𝑋→2
(1 + 3𝑥)
2
x 1.7 1.8 1.9 1.99 2.01 2.05 2.1 2.2
y 6.1 6.4 6.7 6.97 7.03 7.15 7.3 7.6
L = 7
5b

6. Graph:

7. Instruction: Find the limits and complete the table.
1. 𝐥𝐢𝐦
𝒙→𝟑
𝟑𝒙 + 𝟓
3
X 2.93 2.95 2.99 3.03 3. 1 3.3
y
L =
2. 𝐥𝐢𝐦
𝒙→𝟒
𝒙 𝟐
− 𝟏𝟎
4
X 3.75 3.85 3.95 4.05 4. 15 4.25
y
L =
3. 𝐥𝐢𝐦
𝒙→𝟐
𝒙 𝟐
+ 𝟑𝒙 − 𝟔
2
X 2. 96 2.97 2. 98 2.99 3.01 3.02 3.03 3.04
y
L =
4. 𝐥𝐢𝐦
𝒙→𝟓
(𝒙 − 𝟐) 𝟐
5
X 4.90 2.94 4. 98 5.03 5.07 5.1
y
L =
5. 𝐥𝐢𝐦
𝒙→𝟐
(𝒙 + 𝟕𝒙 − 𝟏) 𝟑
5
X 1.89 1.93 1.97 1.99 2. 02 2.06 2.1 2.4
Y
L =
Activity 2: You Complete
Me!!!
Activity 3: Guess
Where I Am??
6b

8. Instruction: Graphs the following.
1. 𝐥𝐢𝐦
𝒙→−𝟏
𝒙 𝟐
− 𝟒
−1
x -1.07 -1.05 -1.01 -0.99 -0.97 -0.95
y -2.86 -2.90 - 2.98 - 3.02 - 3.06 -3.1
L= -3
2. 𝐥𝐢𝐦
𝒙→−𝟑
𝒙 𝟐
− 𝟐𝒙 + 𝟏
3
x 2.93 2.95 2.97 2.99 3.01 3.03 3.05 3.07
y 3.72 3.80 3.88 3.96 4.04 4.12 4.20 4.28
L= 4
3. 𝐥𝐢𝐦
𝒙→𝟒
𝒙 𝟐
− 𝟐𝟓
4
x 3.7 3.8 3.9 4.02 4.05 4.07
y -11.31 -10.56 -9.79 --8.84 -8.60 -8.44
L= -9
7b

9. 8b
Solve the following limits using the table of values and using the graph.
1. lim
𝑥→3
(1 + 4𝑥)
3
X 2.75 2.88 2.9 2.98 3.02 3.1 3.12 3.15
y
L=
2. lim
𝑥→4
𝑥2
− 3𝑥 + 2
4
X 3.88 3.9 3.93 3.98 4.02 4.1 4.13 4.15
y
L=
Activity 3: Lets Get
Harder….Baby!!

10. 10
Lesson 1 showed us how limits can be determined through either a table of
values or the graph of a function. One might ask: Must one always construct a
table or graph the function to determine a limit? Filling in a table of values
sometimes requires very tedious calculations. Likewise, a graph may be difficult to
sketch. However, these should not be reasons for a student to fail to determine a
limit.
In this lesson, we will learn how to compute the limit of a function using Limit
Theorems.
Teaching Tip:
It would be good to recall the parts of Lesson 1 where the students were
asked to give the value of a limit, without aid of a table or a graph. Those
exercises were intended to lead to the Limit Theorems. These theorems are a
formalization of what they had intuitively concluded then.
Limit Theorem:
1. Identity Function
The limitof x as x approaches c is equalto c.
This maybe thought of as the substitution law, because x is simply
substituted by c.
lim
𝑥→𝑐
𝑥 = 𝑐
Examples:
1. lim
𝑥→−8
𝑥 = −8
2. lim
𝑥→25
𝑥 = 25
3. lim
𝑥→4
𝑥 = 4
2. Constant Rule
The limit of a constant is itself. If k is any constant, then,
lim
𝑥→𝑐
𝑘 = 𝑘
Examples:
Lesson 1.B. Limit Theorems
9
INTRODUCTION

11. 11
1. lim
𝑥→−8
10 = 10
2. lim
𝑥→25
18 = 18
3. lim
𝑥→4
−9 = −9
3. Sum Rule
This says that the limit of a sum of functions is the sum of the
limits of the individual functions.
lim
𝑥→𝑐
(𝑓( 𝑥)) + (𝑔( 𝑥)) = lim
𝑥→𝑐
𝑓( 𝑥) + lim
𝑥→𝑐
𝑔(𝑥)
Examples:
1. lim
𝑥→2
( 𝑥 + 4) = lim
𝑥→2
𝑥 + lim
𝑥→2
4
= 2 + 4
= 6
2. lim
𝑥→5
( 𝑥 + 10) = lim
𝑥→5
𝑥 + lim
𝑥→5
10
= 5 + 10
= 15
4. Difference Rule
The limit of a difference of functions is the difference of their
limits.
lim
𝑥→𝑐
(𝑓( 𝑥)) − (𝑔( 𝑥)) = lim
𝑥→𝑐
𝑓( 𝑥) − lim
𝑥→𝑐
𝑔(𝑥)
Examples:
1. lim
𝑥→8
( 𝑥 − 6) = lim
𝑥→8
𝑥 − lim
𝑥→8
6
= 8 - 6
= 2
2. lim
𝑥→3
( 𝑥 − 10) = lim
𝑥→3
𝑥 − lim
𝑥→3
10
= 3 – 10
= -7
5. Product of a Constant and Function
The constant k can be factored outside if a limit.
lim
𝑥→𝑐
𝑘. 𝑓(𝑥)
Examples:
1. lim
𝑥→8
4𝑥 = 4 . lim
𝑥→8
𝑥
= 4 . 8
= 32
2. lim
𝑥→2
10𝑥 = 10 . lim
𝑥→2
𝑥
= 10 . 2
= 20
10b

12. Instruction: Solve the following limits.
1. lim
𝑥→−9
−13 6. lim
𝑥→−226
2𝑥
2. lim
𝑥→111
𝑥 7. lim
𝑥→41
(𝑥 + 23)
3. lim
𝑥→14
(𝑥 + 30) 8. lim
𝑥→38
(𝑥 + 15 − 6)
4. lim
𝑥→−4
( 𝑥 − 3) 9. lim
𝑥→7
(𝑥 − 9 + 16)
5. lim
𝑥→67
73 10. lim
𝑥→18
7𝑥
Continuation……………………………..
6. Product Rule
The limits of a product of two quantities of their limits.
lim
𝑥→𝑐
(𝑓( 𝑥). 𝑔(𝑥)) = lim
𝑥→𝑐
𝑓( 𝑥).lim
𝑥→𝑐
𝑔(𝑥)
Examples:
1. lim
𝑥→−1
𝑥 (𝑥 − 11) = lim
𝑥→−1
𝑥 ( lim
𝑥→−1
𝑥 − lim
𝑥→−1
11)
= -1 (-1-11)
= -1 (-12)
= 12
2. lim
𝑥→2
𝑥 (𝑥 + 3) = lim
𝑥→2
𝑥 (lim
𝑥→2
𝑥 + lim
𝑥→2
3)
= 2 (2 + 3)
=2 (5)
= 10
7. Quotient Rule
The limits of the two functionis the quotient of their limit except if
the limit of the denominator is zero.
Lim
𝑥→𝑐
𝑓( 𝑥)
𝑔( 𝑥)
=
lim
𝑥→𝑐
𝑓( 𝑥)
lim
𝑥→𝑐
𝑔( 𝑥)
Examples:
1. Lim
𝑥→−1
𝑥
(𝑥−4)
=
lim
𝑥→−1
𝑥
lim
𝑥→−1
𝑥− lim
𝑥→−1
4
=
−1
−1−4
Activity 1. LOTS
At this point, you are going to solve the first 5 rules of limit threorem.
11b

13. 13
=
−1
−5
=
1
5
2. Lim
𝑥→2
3𝑥
(𝑥+9)
=
3. lim
𝑥→3
𝑥
lim
𝑥→2
𝑥+ lim
𝑥→2
9
=
3.2
2+9.
=
6
11
8. Power Rule
lim
𝑥→𝑐
𝑥 𝑛
= 𝑐 𝑛
, where n is positive number.
Examples:
1. lim
𝑥→3
𝑥4
= 34
2. lim
𝑥→−7
𝑥5
= (−7)5
= 81 = -16807
9. Power Function Rule
The limit of an 𝑛𝑡ℎ
power of a function is the 𝑛𝑡ℎ
power of the
function, where n is a positive integer.
lim
𝑥→𝑐
𝑓(𝑥) 𝑛
Examples:
1. lim
𝑥→2
(𝑥2
− 2𝑥 + 3)5
= (lim
𝑥→2
𝑥2
− 2.lim
𝑥→2
𝑥 + lim
𝑥→2
3)
= (22
− (2.2) + 3)5
= (4 − 4 + 3)5
= (3)5
= 243
2. lim
𝑥→−1
(2𝑥2
− 5)3
=(2. lim
𝑥→−1
𝑥2
− lim
𝑥→−1
5)
3
= (2.(−1)2
− 5)3
= (2.1 − 5)3
= (−3)3
= -27
10.Radical Rule
lim
𝑥→𝑐
√ 𝑥𝑛
= √ 𝑥𝑛
, where n is a positive integer and c>0 if n is even.
Examples:
1. lim
𝑥→16
√ 𝑥4
= √16
4
2. lim
𝑥→1419857
√ 𝑥5
= √1419857
5
= 2 = 17
11.Radical Function Rule
The limit of the 𝑛𝑡ℎ
power root of the function is the 𝑛𝑡ℎ
root of
the limit of the function, where n is a positive integer, and f(x) > 0 if n is
12b

14. 14
even lim
𝑥→𝑐
√ 𝑓(𝑥)𝑛
= √lim
𝑥→𝑐
𝑓(𝑥)𝑛
wher n is a positive number and f(x)>0 if n is
even.
Example:
𝐥𝐢𝐦
𝒙→−𝟐
√𝒙 𝟒 + 𝟒𝒙 𝟐 − 𝟕𝒙 + 𝟓
𝟑
= √ lim
𝑥→−2
𝑥4 − 4. lim
𝑥→−2
𝑥2 − 7. lim
𝑥→−2
𝑥 + lim
𝑥→−2
53
= √(−2)4 − 4(−2)2 − 7(−2)+ 53
= √16+ 4(4)+ 14 + 5
3
= √16+ 16 + 19
3
= √51
3
Instruction: Solve the following limit theorems.
1. lim
𝑥→9
𝑥 (𝑥 − 31) 6. lim
𝑥→16
𝑥8
2. Lim
𝑥→3
2𝑥
(𝑥+5)
7. lim
𝑥→5
(𝑥3
+ 8𝑥 + 16)4
3. lim
𝑥→−4
(𝑥2
+ 4𝑥 + 4)3
8. lim
𝑥→11
𝑥 (𝑥 − 15)
4. lim
𝑥→10
𝑥4
9. lim
𝑥→−3
√𝑥2 − 7𝑥 + 10
5
5. lim
𝑥→125
√ 𝑥3
10. lim
𝑥→4
√𝑥3 + 3𝑥3 + 6𝑥 + 9
3
Instruction: Solve the following:
Activity 2.
HOTS
In the next activity, you are going to identify what are the theorem used and solve all the
limits theorem.
At this juncture, you are going to solve the remaining rules of limit theorem.
Activity 3. Did you Get
ME??
13b

15. 14b
I. Choose the best answer. Write the letter to provided space before the
number.
A. Product Rule E. Quotient Rule I. Radical Function Rule
B. Radical Rule F. Product of a Constant and Function J. Power Function Rule
C. Difference Rule G. Sum Rule K. Limit of the Identity Function Rule
D. Power rule H. Constant rule
____________1. The constant k can be factored outside of a limit.
____________2. lim
𝑥→𝑐
𝑥 = 𝑐
____________3. The limit of the difference is the difference of their limits.
____________4. The limit of the sum of two function is the sum of their limits.
____________5. lim
𝑥→𝑐
𝑘 = 𝑘
____________6. The limit of a product of two quantities of their limits.
____________7. lim
𝑥→𝑐
𝑥 𝑛 = 𝑐2 , where n is a positive number.
____________8. The limit of an nth power of a function is the nth power of the function,
where n is positive integer.
____________9. The limit of a quotient of two function is the quotient of their limits except
if the limit of the denominator is zero.
____________10. lim
𝑥→𝑐
√ 𝑥 =𝑛
√ 𝑐𝑛
, where n is a positive integer and c > 0 if n is even.
____________11. The limit of the nth power root of a function is the nth root of the limit of
the function, where n is a positive integer, and f(x) > if n is even.
II. Match the column A to column B. Show your solution on your paper.
Column A Column B
1. lim
𝑥→9
𝑥 a. -6
2. lim
𝑥→4
( 𝑥 + 5) b. 25
3. lim
𝑥→−1
(6𝑥4 − 3𝑥3 − 4𝑥2 + 7𝑥 − 5)4 c. 11
4. lim
𝑥→3
7𝑥 d. 2
5. lim
𝑥→1345
29 e. 45
6. lim
𝑥→5
𝑥2 = 52
f.
1
5
7. lim
𝑥→−1
𝑥
𝑥−4
g. 9
8. lim
𝑥→16
√ 𝑥4
h. 2401
9. lim
𝑥→16
√𝑥4 + 4𝑥23
− 7𝑥 + 5 i. √513

16. 10. lim
𝑥→3
𝑥 ( 𝑥 + 12 ) j. 29
11. lim
𝑥→4
( 𝑥 − 10) k. 21
III. Solve the following limits.
1. lim
𝑥→2
15𝑥
2. lim
𝑥→15
( 𝑥 − 3)
3. lim
𝑥→1000000
15
4. lim
𝑥→6
𝑥 ( 𝑥 − 4 )
5. lim
𝑥→3
7𝑥
𝑥+4

17. Limit at Infinity and Infinite Limit aims to prepare and nurture the students for
them to become globally competitive with this objectives , the author has come
up with this rich material which is designed as a function prerequisite for research
and other high mathematics subjects.
INFINITE LIMIT THEOREM 1
Let n by any positive integer, then
a. lim
𝑥→0+
1
𝑥 𝑛
= +∞
b. lim
𝑥→0−
1
𝑥 𝑛
= +∞ if n is odd
= -∞ if n is even
Examples:
a. lim
𝑥→0+
1
𝑥3
= +∞
b. lim
𝑥→0−
1
𝑥10
= +∞
c. lim
𝑥→0−
1
𝑥7
= -∞
INFINITE LIMIT THEOREM 2
Let c be any real number. Suppose in lim
𝑥→𝑐
𝑓(𝑥)
𝑔(𝑥)
, lim
𝑥→𝑐
𝑔( 𝑥) = 0 , lim
𝑥→𝑐
𝑓(𝑥) =
𝑘, where k is a real number and k≠ 0.
Lesson 1. C. Infinite Limit and Limit of Infinity
15b
INTRODUCTON
INFINITE LIMIT

18. a. If k>0 and g(x) → 0 through positive values, then lim
𝑥→𝑐
𝑓( 𝑥)
𝑔(𝑥)
= +∞
b. If k>0 and g(x) → 0 through negative values, then lim
𝑥→𝑐
𝑓( 𝑥)
𝑔(𝑥)
= -∞
c. If k<0 and g(x) → 0 through positive values, then lim
𝑥→𝑐
𝑓( 𝑥)
𝑔(𝑥)
= -∞
d. If k<0 and g(x) → 0 through negative values, then lim
𝑥→𝑐
𝑓( 𝑥)
𝑔(𝑥)
= +∞
Summary:
through k answer
+ >0 +∞
- >0 -∞
+ <0 -∞
- <0 +∞
Examples:
a. lim
𝑥→4+
4𝑥+1
4−𝑥
=
4 (4)+1
4−4
b. lim
𝑥→3−
𝑥2
−2𝑥+4
𝑥−3
=
(3)2
−2(3)+4
3−3
=
16+1
0
=
9−6+4
0
=
17
0
=
7
0
k = 17 = k = 7
17 > 0 = 7 > 0
= +∞ = -∞
𝑐. lim
𝑥→−2+
2𝑥−1
𝑥+2
=
2 (−2)−1
−2+2
b. lim
𝑥→−2−
2𝑥−5
𝑥+2
=
2 (−2)−5
−2+2
=
−4−1
0
=
−4−5
0
=
−5
0
=
−9
0
k = -5 k= -9
-5 < 0 -9 >0
= -∞ = +∞
Solve the following limitsa and identify what are the theorems used.
At this point, you are going to solve the the infinite limit theorem before we proceed
to the next lesson.
16b
Activity 1. INFINITE
LIMIT

19. 1. lim
𝑥→0+
1
𝑥2
_________________
2. lim
𝑥→0+
1
𝑥10 ____________________
3. lim
𝑥→0−
1
𝑥36 ____________________
4. lim
𝑥→−3+
𝑥2
−5𝑥+3
3𝑥+9
__________________
5. lim
𝑥→5
3𝑥−6
𝑥−5
__________________
6. lim
𝑥→5
13𝑥−7
𝑥+4
__________________
Limit at Infinity Theorem 1
Let r be any positive integer, then
a. lim
𝑥→+∞
1
𝑥 𝑟
= 0
b. lim
𝑥→−∞
1
𝑛 𝑟
= 0
Examples:
a. lim
𝑥→+∞
1
𝑥4
= 0
b. lim
𝑥→−∞
1
𝑥5
= 0
Limit at Infinity Theorem 2
Let n be a positive real number and k any real number except 0, then
Formula Answer
a.
lim
𝑥→+∞
𝑘
𝑛 𝑟
= 0
b.
lim
𝑥→−∞
𝑘
𝑛 𝑟
=0
c lim
𝑥→+∞
𝑘 . 𝑥 𝑛
= +∞ if k > 0
= −∞ if k < 0
LIMIT OF INFINITY
17b

20. d. lim
𝑥→−∞
𝑘 . 𝑥 𝑛
= +∞ if k > 0 and n
is even or k < 0 andn
is odd.
= −∞ if k > 0 and n
is odd or K < o and n
is even.
Reminder for limit at infinity theorem 2.D
Value of K n Answer
> Even +∞
> Odd −∞
< Even −∞
< Odd +∞
Examples:
a. lim
𝑥→+∞
2
3𝑥3 = 0 b. lim
𝑥→−∞
1
𝑥4 = 0
𝑐. lim
𝑥→+∞
5𝑥4
= k = 5 lim
𝑥→+∞
−2𝑥4
= k = -2
5 > 0 -2 < 0
=+∞ =−∞
𝑑. lim
x→−∞
4𝑥2
= k = 4 lim
x→−∞
8𝑥3
= k = 8
4 > 0 8 > 0
n = even n = odd
=+∞ =−∞
lim
x→−∞
−11𝑥4
= k = -11 lim
x→−∞
−9𝑥17
= k = -9
-11 < 0 -9 < 0
n =even n = odd
= −∞ =+∞
Limit at Infinity Theorem 3
Let h(x) =
𝑓(𝑥)
𝑔(𝑥)
, where f(x) and g(x) are polynomials.
18b

21. a. If the degree of f (x) is less than the degree of g(x), then
lim
𝑥→+∞
𝑓(𝑥)
𝑔(𝑥)
= 0 lim
𝑥→−∞
𝑓(𝑥)
𝑔(𝑥)
= 0
N < D = 0
Examples:
lim
𝑥→+∞
3𝑥4
−5𝑥3
+2𝑥
𝑥5+4𝑥−7
lim
𝑥→−∞
7𝑥2
−𝑥3
+2
𝑥4+3𝑥−9
n = 4 n = 2
d = 5 d = 4
4 < 5 2 < 4
= 0 = 0
b. If the degree of f(x) is equal to the degree of g(x), a is the leading
coefficient of f(x) and b is the leading coefficient of g(x), then
lim
𝑥→+∞
𝑓(𝑥)
𝑔(𝑥)
=
𝑎
𝑏
lim
𝑥→−∞
𝑓(𝑥)
𝑔(𝑥)
=
𝑎
𝑏
n = d  coefficient =
𝑎
𝑏
Examples:
lim
𝑥→+∞
𝑥3
+2𝑥
2𝑥3 +3𝑥−4
lim
𝑥→−∞
6𝑥5
+2𝑥
5𝑥5 +3𝑥−4
n = 3 n = 5
d = 3 d = 5
3 = 3 5 = 5
=
1
2
=
6
5
c. If the degree of f (x) is greater than g(x), then
lim
𝑥→+∞
𝑓(𝑥)
𝑔(𝑥)
lim
𝑥→−∞
𝑓(𝑥)
𝑔(𝑥)
Examples:
lim
𝑥→+∞
𝑥4
−16
𝑥2−4𝑥+4
lim
𝑥→−∞
𝑥3
+3𝑥−12
𝑥2−7𝑥+12
n = 4 n = 3
d = 2 d = 2
=+∞ =−∞
19b
Activity 2. LIMIT AT
INFINITY

22. Answer the following problems by using the Limit at Infinity Theorems.
1. lim
𝑥→+∞
1
𝑥4 = _____
2. lim
𝑥→−∞
1
𝑥6 = _____
3. lim
𝑥→+∞
(
1
𝑥5) = _____
4. lim
𝑥→−∞
(
3
𝑥7) = _____
5. lim
𝑥→−∞
(
𝑥3
−1
𝑥2−25
) = _____
I. Answer the following infinite limite and limit at infinity and Identify
what are the theorems used.
1. lim
𝑥→+∞
𝑥2
+6𝑥
2𝑥7 +3𝑥−10
2. lim
𝑥→5+
9𝑥+4
5−𝑥
3. lim
𝑥→0−
1
𝑥3
4. lim
𝑥→+∞
−12𝑥7
5. lim
𝑥→+∞
9𝑥3
−8𝑥2
+7𝑥
𝑥4+4𝑥−3
6. lim
𝑥→0−
1
𝑥17
7. lim
x→−∞
2𝑥5
8. lim
𝑥→−3+
2𝑥−1
𝑥+2
9. lim
𝑥→5
3𝑥−6
𝑥−5
10. lim
𝑥→+∞
4𝑥4
−5𝑥3
+2𝑥
2𝑥4 +4𝑥−7
II. Complete the table below.
1. lim
𝑥→−∞
𝑘 . 𝑥 𝑛
Value of K n Answer
> Even Type equation here.
> Odd Type equation here.
< Even Type equation here.
< Odd Type equation here.
2. INFINITE LIMIT THEOREM 2
ACTIVITY 3. I WANT
MORE!
20b

23. through k answer
+ >0 +∞
- >0 -∞
+ <0 -∞
- <0 +∞
INSTRUCTION: Solve the following show your solution on the table of values.
Show the graph.
1. lim
𝑥→2
(𝑥 + 10𝑥)
2
x 1.5 1.6 1.7 1.9 2.5 2.6 2.7 2.9
y
2. lim
𝑥→4
(2 + 6𝑥)
4
x 3.15 3.20 3.25 3.30 4.15 4.20 4.25 4.30
y
INSTRUCTION: Solve the following limit theorems.
1. lim
𝑥→9
𝑥 (𝑥 − 31) 6. lim
𝑥→16
𝑥8
2. Lim
𝑥→3
2𝑥
(𝑥+5)
7. lim
𝑥→5
(𝑥3
+ 8𝑥 + 16)4
3. lim
𝑥→−4
(𝑥2
+ 4𝑥 + 4)3
8. lim
𝑥→11
𝑥 (𝑥 − 15)
4. lim
𝑥→10
𝑥4
9. lim
𝑥→−3
√𝑥2 − 7𝑥 + 10
5
5. lim
𝑥→125
√ 𝑥3
10. lim
𝑥→4
√𝑥3 + 3𝑥3 + 6𝑥 + 9
3
INSTRUCTION: Solve the following problems.
1. lim
𝑥→+∞
2
3𝑥3
POST-TEST
21

24. 2. lim
𝑥→−∞
1
𝑥4 = 0
3. 𝑐. lim
𝑥→+∞
5𝑥4
4. lim
𝑥→+∞
−2𝑥4
5. 𝑑. lim
x→−∞
4𝑥2
6. lim
x→−∞
8𝑥3
7. lim
x→−∞
−11𝑥4
8. lim
x→−∞
−9𝑥17