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BASIC CALCULUS:
Taking on Limits and
Continuity
11th Grade

2. What are the fundamental principles and
techniques used to understand and
evaluate limits in basic calculus, and how
do they apply to real-world problems and
mathematical analysis?
ESSENTIAL
QUESTION

3. Is the study of Calculus
really necessary?

4. YE
S
It’s fundamental for fields like the
ff:
● Physics
● Engineering
● Economics
● Biology
● Computer Science
It provides a foundation for
understanding how things
change over time.

5. It is the backbone of calculus
In mathematics, it formally expresses the concept
of arbitrary closeness. That is, a limit is a value
that a variable approaches as closely as one
desires.
What role does Limits
play in Calculus?
It allows us to understand Calculus’ two
mathematical operations, differentiation and
integration.

6. Table of contents (Limits: Part 1)
01
Illustrating the Limit of a Function using Table of
Values
02
Definition of a Limit, One-sided and Two-sided
Limit
03 Illustrating the Limit of a Function using Graphs
04 Theorems on Limits of Functions
05 Limits at Infinity

7. FUNCTION RECALL
Evaluate the following functions at a specific value of x.
1. 𝑓 𝑥 = 2𝑥2 − 5𝑥 + 9; 𝑥 = 3
2. 𝑓 𝑥 = 4𝑥 − 7 ; 𝑥 = −3
3. 𝑓 𝑥 = 𝑐𝑜𝑠 2𝑥 ; 𝑥 =
𝜋
4
4. 𝑓 𝑥 = 𝑙𝑜𝑔2 𝑥 ; 𝑥 = 8
5. f 𝑥 =
2 𝑥 − 5
(2𝑥3+𝑥2−10)
; 𝑥 = 4

8. Illustrating the
Limit of a
Function
using Table of
Values
f (x)
1
x
2
3
4
-1
1 2 3 4 5 6 7 8

9. The limit of 4x + 5 as x approaches to 1 is
equal to 9 can be written as lim
𝑥→1
(4𝑥 +

10. x 0.9 0.99 0.999 0.9999
f(x) = 4x + 5 8.6 8.96 8.996 8.9996
Let us take values of x lower than 1 but very close to 1 and
substitute these values to 4x + 5.

11. x 1.1 1.01 1.001 1.0001
f(x) = 4x + 5 9.4 9.04 9.004 9.0004
Let us take values of x higher than 1 but very close to 1 and
substitute these values to 4x + 5.

12. x 1.1 1.01 1.001 1.0001
f(x) = 4x + 5 9.4 9.04 9.004 9.0004
x 0.9 0.99 0.999 0.9999
f(x) = 4x + 5 8.6 8.96 8.996 8.9996
values of x lower than 1 but very close to 1
values of x higher than 1 but very close to 1

13. Illustrating the
Limit of a
Function
using Graphs
f (x)
1
x
2
3
4
-1
1 2 3 4 5 6 7 8

18. Limits in maths
f (x)
y
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Limits
lim f(x) = A
x→a
→ f(x) is a function
→ x is a variable
approaching to value a
Limit formula

19. Activity 01: true or false
True False
Explication
Every continuous function is always differentiable
Say true or false and write an explanation in your own words

20. Mercury is the closest planet
to the Sun and the smallest
of them all
Limit of a function
Venus has a beautiful name
and is the second planet
from the Sun
Continuity of a function
Despite being red, Mars is
actually a cold place. It’s full
of iron oxide dust
One-sided limits
Limits and functions

21. What are functions?
Venus has a beautiful name and is
the second planet from the Sun
Saturn is the second-largest planet in
the Solar System
Function notation
Function operations
Mercury is the closest planet to the
Sun and the smallest of them all
Despite being red, Mars is actually a
cold place. It’s full of iron oxide dust
Function definition
Function graphs

22. Properties of limits
Law of addition lim [f(x)+g(x)]=lim f(x)+lim g(x)
x→a x→a x→a
Law of subtraction
Law of multiplication
Law of division
Law of power lim c=c
x→a
lim [f(x)−g(x)]=lim f(x)−lim g(x)
x→a x→a x→a
lim [f(x)⋅g(x)]=lim f(x)⋅lim g(x)
x→a x→a x→a
lim [ ]= ,where lim g(x) ≠ 0
x→a
f(x)
g(x)
lim f(x)
x→a
lim g(x)
x→a
x→a

23. Definition of continuity
Venus has a beautiful name,
but also high temperatures
Jupiter is a gas giant and
has around eighty moons
No holes or gaps
Nearby points are close
Mercury is the smallest
planet of them all
Despite being red, Mars is
actually a very cold place
Uninterrupted behavior
Graph is connected
Earth is the third planet from
the Sun and has life
Saturn is composed of
hydrogen and helium
Limits exist
No vertical asymptotes

24. Special
rules
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2 3 4 5 6

25. “This is a quote, words
full of wisdom that
someone important
said and can make the
reader get inspired”
—Someone Famous
f (x)
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26. A picture is worth a thousand words

27. To reinforce the
concept, try using
an image
Images reveal large amounts of data,
so remember: use an image instead of
a long text. Your audience will
appreciate it

28. Activity 03: matching
Matching concepts with description
… are defined as the values that a function approaches
the output for the given input values
… is a property of functions characterized by the
absence of abrupt changes or breaks in their graphs
… is visualized as a rule, which gives a unique output
for every input x
Limits
Continuity
Function

29. 150,000
Big numbers catch your audience’s attention

30. 9h 55m 23s
Jupiter’s rotation period
386,000 km
Distance between Earth and the Moon
333,000
The Sun’s mass compared to Earth’s

31. Some percentages
Mercury is the closest planet to the Sun and
the smallest of them all
About Mercury
Venus has a beautiful name and is the
second planet from the Sun
About Venus
20%
0% 100%
80%
0% 100%

32. Limits of functions and continuity
Limits of functions
Mercury is the closest planet to the Sun and
the smallest of them all
Venus has a beautiful name and is the
second planet from the Sun
Continuity
Jupiter is a gas giant and the biggest planet
in the Solar System
Saturn is the second-largest planet
in the Solar System
Despite being red, Mars is actually a cold
place. It’s full of iron oxide dust
Neptune is the fourth-largest planet
in the Solar System

33. Activity 04: complete
A 1
B 0
C ∞
D does not exist
lim sin x equals
x→∞
Mark the right answer with colour

34. Computer mockup
You can replace the image on the screen with
your own work. Just right-click on it and select
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35. Tablet mockup
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36. Phone mockup
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37. This is a map
Earth
Saturn
Mercury
Jupiter
Mars
Neptune
Venus
Description Description Description

38. Techniques for solving limit problems
Mercury is the
smallest planet of
them all
Venus is extremely
hot, even more than
Mercury
Earth is the only
planet known to
harbor life
Despite being red,
Mars is actually a
cold place
Jupiter is the
biggest planet in the
Solar System
Saturn has a high
number of moons,
like Jupiter
Neptune is the
eighth planet from
the Sun
The Sun is the star
at the center of the
Solar System

39. Challenges and common mistakes
01 Mercury is the closest planet to the Sun and the smallest of them all
02 Venus has a beautiful name and is the second planet from the Sun
03 Despite being red, Mars is actually a cold place. It’s full of iron oxide dust
04 Jupiter is a gas giant and the biggest planet in the Solar System
05 Saturn is the second-largest planet in the Solar System
06 Neptune is the fourth-largest planet in the Solar System

40. Key takeaways
Conclusion 01 Conclusion 02 Conclusion 03
Venus has a beautiful name
and is the second planet from
the Sun. It’s terribly hot, even
hotter than Mercury, and its
atmosphere is extremely
poisonous
Jupiter is a gas giant and the
biggest planet in the Solar
System. It’s the fourth-
brightest object in the night
sky. It was named after the
Roman god of the skies and
lightning
Saturn is a gas giant and has
several rings. It’s composed
mostly of hydrogen and
helium. It was named after the
Roman god of wealth and
agriculture

41. Understanding limits and continuity
Follow the link in the graph to modify its data and then paste
the new one here. For more info, click here
Mercury is the closest planet to the Sun and
the smallest of them all
Venus has a beautiful name and is the
second planet from the Sun. It’s terribly hot,
even hotter than Mercury, and its
atmosphere is extremely poisonous
Limits and continuity
𝒇

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