This document appears to be a series of slides from a calculus lesson discussing continuity of functions. It includes definitions of key concepts like continuity and discontinuity. There are examples of determining whether functions are continuous or discontinuous at given points by analyzing limits. It also discusses real-world applications and asks students to solve practice problems to understand the material. Interactive elements like group discussions and games are used to engage students in applying the concepts.

1. SLIDESMANIA
c o 1 Q3 Calculus

2. SLIDESMANIA
Grade 12 – General Academic Strand B
Basic Calculus
JOEYNEL L. PONTEVEDRA, MST

3. SLIDESMANIA
Revisiting
classroom house
rules.
Class secretary, kindly check
the attendance.
Everyone is
requested to
stand for a prayer.
Greetings…

4. SLIDESMANIA
Find the limit of the given function below.
A. lim
𝑥 →−3
𝑓 𝑥
B. lim
𝑥 →−1
𝑓 𝑥
C. lim
𝑥 →1
𝑓 𝑥
D.lim
𝑥 →3
𝑓 𝑥
E. lim
𝑥 →5
𝑓 𝑥
= DNE
= DNE
= -1
= DNE
= 5

5. SLIDESMANIA
1 0 1 1 1

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1 1 0 0 1

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1 1 1 1 1

8. SLIDESMANIA
Define
continuity of
the function;
Determine
whether a
function is
continuous or
not at a given
number or
point;
Cite real-life
applications of
continuity and
discontinuity of
a function at a
point and
provide its
important
contributions.
Learning Objectives

9. SLIDESMANIA
Define continuity of the function;

10. SLIDESMANIA
Determine whether a function is
continuous or not at a given
number or point;

11. SLIDESMANIA
Cite real-life applications of continuity and
discontinuity of a function at a point and
provide its important contributions.

12. SLIDESMANIA
Make sure to watch the
video keenly and observe
the notable events.
Questions will follow after
the video presentation.
Let’s watch a video.

13. SLIDESMANIA
Based on the video
presented, can you please
share to the class your
observation.

14. SLIDESMANIA
At some point, how
significant is discontinuity
in our life?

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Cite other example of
continuity that’s beneficial
to your life as a student.

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SLIDESMANIA
Trace the drawing
without lifting your
hand.
Start here…
END.

17. SLIDESMANIA
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Trace the Graph
Is the function continuous at x=1?
Start here…
END.

18. SLIDESMANIA
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Trace the Graph
CONTINUOUS

19. SLIDESMANIA
SLIDESMANIA
Trace the Graph
Is the function continuous at x=1?
Start here…

20. SLIDESMANIA
SLIDESMANIA
Trace the Graph
DISCONTINUOUS

21. SLIDESMANIA
SLIDESMANIA
Trace the Graph
Is the function continuous at x=0?
Start here…
END.

22. SLIDESMANIA
SLIDESMANIA
Trace the Graph
DISCONTINUOUS

23. SLIDESMANIA
Suppose we are not given the graph of a function
but just the function itself. How do we determine if
the function is continuous at a given number?

24. SLIDESMANIA
Please proceed to your
groupmates and discuss
how the problems are
solved. Answer the
questions that follows.
Accomplish this in only
10minutes.

25. SLIDESMANIA
Example 4:
Determine if
𝑓 𝑥 = 𝑥3
+ 𝑥2
− 2
is continuous or not at x = 1.
Solution:
We have to check the three conditions
for continuity of a function.
a. 𝐼𝑓 𝑥 = 1 , 𝑡ℎ𝑒𝑛 𝑓 1 = 0.
b. lim
𝑥 →1
𝑓 𝑥 = lim
𝑥 →1
(𝑥3 + 𝑥2 − 2)
= 13
+ 12
− 2
= 0
a. 𝑓 1 = 0 = lim
𝑥 →1
𝑓 𝑥
Therefore, f is continuous at x = 1.

26. SLIDESMANIA
Example 5:
Determine if
𝑓 𝑥 =
𝑥2 − 𝑥 − 2
𝑥 − 2
is continuous or not at x = 0.
Solution:
We have to check the three conditions
for continuity of a function.
a. 𝐼𝑓 𝑥 = 0 , 𝑡ℎ𝑒𝑛 𝑓 0 = 1.
b. lim
𝑥 →0
𝑓 𝑥 = lim
𝑥 →0
𝑥2 − 𝑥 − 2
𝑥 − 2
= lim
𝑥 →0
𝑥−2 (𝑥+1)
𝑥 − 2
= lim
𝑥 →0
(𝑥 − 1)
= 1
c. 𝑓 0 = 1 = lim
𝑥 →0
𝑓(𝑥) .

27. SLIDESMANIA
Example 6:
Determine if
𝑓 𝑥 =
𝑥2 − 𝑥 − 2
𝑥 − 2
is continuous or not at x = 2.
Solution:
We have to check the three conditions for
continuity of a function.
a. 𝐼𝑓 𝑥 = 2 , 𝑡ℎ𝑒𝑛
𝑓 2 = 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑.
a. lim
𝑥 →2
𝑓 𝑥 = lim
𝑥 →0
𝑥2 − 𝑥 − 2
𝑥 − 2
= lim
𝑥 →2
𝑥−2 (𝑥+1)
𝑥 − 2
= lim
𝑥 →2
(𝑥 − 1)
= 1
c. 𝑓 2 = 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑 ≠ lim
𝑥 →2
𝑓(𝑥) .

28. SLIDESMANIA
The game is called AUCTION. The teacher will post a question, group
leaders will then bid certain amount to have a chance in answering the
question. The highest bidder will answer first the question, if the answer
is correct, all the amount bid will be given to them. However, if the
answer is wrong the next bidder will have a chance to steal.

29. SLIDESMANIA
Question No. 1:
Determine if
𝑓 𝑥 =
𝑥2
− 𝑥 − 2
𝑥 + 1
is continuous or not at x = -1.
NOT
CONTINUOUS

30. SLIDESMANIA
Question No. 2
1. The following are the conditions on
determining if a function is continuous or
not.
a. 𝑓 𝑐 𝑒𝑥𝑖𝑠𝑡
b. lim
𝑥 →3
𝑓 𝑥 𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑒𝑥𝑖𝑠𝑡; &
𝑐. 𝑓 𝑐 ≠ lim
𝑥 →3
𝑓 𝑥
Which of the steps is wrong?
A. A only C. B and C
B. A and B D. No mistake
C

31. SLIDESMANIA
Question No. 3
The following examples are all related to
continuity. Which of these examples is
harmful to human beings?
A. The circular flow of economy
B. The continuous rainfall for 5 days
C. The earth’s revolution around the sun
D. The human brain’s capacity to store
and process information
B

32. SLIDESMANIA
At this time let us know if you really understand our discussion.
Please answer the questions on the test paper provided
by shading the letter of the correct answer.

33. SLIDESMANIA
For Verbal/Linguistics Intelligences:
(May it be English or Filipino)
Create an essay in at most 10 sentences only.
Please refer to the situation given below:
“In the recent trend, SOCIAL MEDIA has been
a channel for the continuous thread of
information. As a responsible student, how will
you filter these information as to important or
not important in your daily living?”

34. SLIDESMANIA
For Musical/Bodily Kinesthetic Intelligences:
Create a group with 5 or more members.
Your task is to choreograph a 5-minute non-stop dance.

35. SLIDESMANIA
For Logical/Mathematical Intelligences:
Solve the problems in a printed material.
Use the FUNCTION CONTINUITY CALCULATOR or
the Symbolab APP in solving the problem.

36. SLIDESMANIA
For Verbal/Linguistics
Intelligences:
(English or Filipino)

37. SLIDESMANIA
For Musical/Bodily
Kinesthetic Intelligences:

38. SLIDESMANIA
THANK YOU and GODBLESS.
JOEYNEL L. PONTEVEDRA, MST