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1. UNIT I
LIMITS AND CONTINUITY

2. CALCULUS
You live in a world that is constantly changing. You
experience things that move at different paces and at
different times. Some are in constant motion, while the
rest are still. Humans have developed several machines
and tools to explain these phenomena. One of the tools
used in explaining motion is calculus.
Calculus is a numerical language that creates
mathematical models to solve problems dealing with
objects or things in motion.

3. Introduction to the Limits
Of Functions

4. LEARNING OBJECTIVES
At the end of this module, I can:
• illustrate the limit of a function using the table of values and
the graph of the function.
• distinguish between lim
𝑥→𝑐
𝑓(𝑥) and 𝑓 𝑐 .
• illustrate the limit laws.
• apply the limit laws in evaluating the limits of algebraic
functions (polynomial, rational, and radical)
• compute limits using a variety of techniques.

5. PRE-TEST

6. PRE-TEST
1. The limit of a function is the value that f(x) gets closer to as x
approaches some number.
2. A two-sided limit only considers values of a function that approaches a
value from either above or below.
3. A two-sided limit 𝒍𝒊𝒎𝒙 → 𝒂𝒇(𝒙) = 𝑳 exists if the one-sided limits and
are the same.
4. Limits by factoring refers to a technique for evaluating limits that
requires finding and eliminating common factors.
5. Evaluating limits by factoring refers to the idea that under certain
circumstances (namely if the function we are examining is continuous),
we can evaluate the limit by simply evaluating the function at the point
we are interested in.

7. INTRODUCTION
• Calculus primarily deals with the limit process as it quantifies the
relationship between two variables or quantities.
• The concept of the limit process eventually leads to the study of
calculus.
• As you have learned in General Mathematics, functions appear in
different forms and they can be represented in many ways. You
may write them in as equations and represent them in words, in
tables of values, and probably in graphical forms.
• The behavior of functions can also be described using the concept
of limits. The limit of a function refers to the value that a “function”
approaches as x approaches a specific value.

8. PRE-
ASSESSMENT

9. LIMIT OF A FUNCTION
• To understand this concept, consider the function defined
by 𝑓 𝑥 =
𝑥−4
𝑥2−16
. By substitution, 𝑓 0 =
0−4
02−16
=
1
4
• Also, as the values of x come close to 0, then the values of
f(x) come close to
1
4
(or 0.25). This is shown in the next
table.

10. LOOKING AT A TABLE OF VALUES
• x approaches 0 from right
𝒇 𝒙 =
𝒙 − 𝟒
𝒙𝟐 − 𝟏𝟔
• x approaches 0 from the left.
x -0.1 -0.01 -0.001 -0.0001 0 0.0001 0.001 0.01 0.1
f(x) 0.2564103 0.250627 0.250063 0.250006 0.25 0.24999 0.249938 0.249377 0.243902
• The case is different when 𝑥 = 4. By substitution, 𝑓(4) does not exist, because the denominator is 0.
However, when the values of x come close to 4, the values of 𝑓(𝑥) come close to 1/8 (or 0.0125).
• x approaches 4 from the left. • x approaches 4 from the right.
x 3.9 3.99 3.999 3.9999 4 4.0001 4.001 4.01 4.1
f(x) 0.12658 0.12516 0.12502 0.125001 0.125 0.124998 0.12498 0.12484 0.12346
𝐥𝐢𝐦
𝒙→𝟎
𝒇 𝒙 = 𝟎. 𝟐𝟓
𝐥𝐢𝐦
𝒙→𝟒
𝒇 𝒙 = 𝟎. 𝟏𝟐𝟓

11. Give two factors in your life
that you consider as limits to
reaching your dreams. How
does each factor affect your
plan of action?

12. What is Limit?

13. LIMIT OF A FUNCTION
•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
lim
𝒙→𝒄
𝒇 𝒙 = 𝑳
This is read, ‘‘The limit of f(x) as x approaches c is L.”

14. Example 1: Evaluate using a table of values
• Approach 2 from its right or values greater
than 2 but close to 2.
lim
𝒙→𝟐
(𝟏 + 𝟑𝒙)
• Approach 2 from its left or values less than 2
but close to 2.

15. LOOKING AT THE GRAPH
lim
𝒙→𝟐
𝟏 + 𝟑𝒙 = 𝟕

16. LIMIT OF A FUNCTION
•Observe that as the values of x get closer and closer to 2,
the values of f(x) get closer and closer to 7. This
behavior can be shown no matter what set of values, or
what direction, is taken in approaching 2. In symbols,
lim
𝒙→𝟐
(𝟏 + 𝟑𝒙) = 𝟕

17. Example : Investigate
lim
𝒙→𝟏
𝒙𝟐
− 𝟓𝒙 + 𝟒
𝒙 − 𝟏
= −𝟑
• Approach 1 from its left • Approach 1 from its right
Take note that 1 is not in the domain of f, but this is not a problem. In evaluating a limit,
remember that we only need to go very close to 1; we will not go to 1 itself.

18. LOOKING AT THE GRAPHS
lim
𝒙→𝟏
𝒙𝟐
− 𝟓𝒙 + 𝟒
𝒙 − 𝟏
= −𝟑

19. ACTIVITY 1

21. The Limit of a Function at c
Vs
The Value of a Function at c
𝒍𝒊𝒎
𝒙→𝟒
𝒇 𝒙 𝒗𝒔 𝒇(𝒄)

22. Is it necessary that
limit of a function as it
approaches c be equal
to its value at c ?

23. •The limit of a function as it approaches c is not
necessarily equal to its value at c. Thus, lim
𝑥→𝑐
𝑓 𝑥
can assume a value different from f(c).
Existence of a Limit
• The limit of a function as x  c exists if
• a. f(c) is defined; or
• b. approaches the same value as x moves closer to c from
both directions.

24. Examples: Evaluate the following limits
Observe that it doesn’t matter if f(2)
is undefined. The function can still
have limit as long as it approaches
the same real number from left and
from the right.
lim
𝒙→𝟐
𝒙𝟐
− 𝟒
𝒙 − 𝟐
x 1.9 1.99 1.999 2 2.001 2.01 2.1
f(x) 3.9 3.99 3.999 ? 4.001 4.01 4.1

25. Examples: Evaluate the following limits
The graph and table of values show that
the limit does not exist. From the left, the
limit as x approaches 2 is -1, while from
the right, the limit as x approaches 2 is 1.
Further, it can be noted that f(2) is not
defined.
lim
𝒙→𝟐
|𝒙 − 𝟐|
𝒙 − 𝟐
x 1.9 1.99 1.999 2 2.001 2.01 2.1
f(x) -1 -1 -1 ? 1 1 1

26. Examples: Evaluate the following limits
From the figure, the graph
approaches 2 as x approaches 2
from both directions. Thus, even if
f(2) = 3, the limit of the function as
x approaches 2 is 2.
lim
𝒙→𝟐
𝒇(𝒙) 𝒘𝒉𝒆𝒏 𝒇 𝒙 =
𝒙, 𝒙 ≠ 𝟐
𝟑, 𝒙 = 𝟐
x 1.9 1.99 1.999 2 2.001 2.01 2.1
f(x) 1.9 1.99 1.999 ? 2.001 2.01 2.1

27. Ex. 4: Investigate through table of values
lim
𝒙→𝟒
𝒇(𝒙) 𝑖𝑓 𝒇 𝒙 =
𝒙 + 𝟏 𝑖𝑓 𝒙 < 𝟒
𝒙 − 𝟒 𝟐
+ 𝟑 𝑖𝑓 𝒙 ≥ 𝟒
• when x approaches 4 from the left, the values
taken should be substituted in f(x) = x + 1
• when x approaches 4 from the right, the
values taken should be substituted in
𝐟(𝐱) = (𝐱 − 𝟒)𝟐+ 𝟑

28. LOOKING AT THE GRAPHS
lim
𝒙→𝟒
𝒇(𝒙) 𝑖𝑓 𝒇 𝒙 =
𝒙 + 𝟏 𝑖𝑓 𝒙 < 𝟒
𝒙 − 𝟒 𝟐 + 𝟑 𝑖𝑓 𝒙 ≥ 𝟒

29. Example 4:
•Observe that the values that f(x) approaches are not
equal, namely, f(x) approaches 5 from the left while it
approaches 3 from the right. In such a case, we say that
the limit of the given function does not exist (DNE). In
symbols,
lim
𝒙→𝟒
𝒇 𝒙 𝑫𝑵𝑬

30. Remark 1:
• We do not say that lim
𝒙→𝟒
𝒇 𝒙 “equals DNE”, nor do we write
“lim
𝒙→𝟒
𝒇 𝒙 = 𝑫𝑵𝑬”, because “DNE” is not a value.
• In the previous example, “DNE” indicated that the function
moves in different directions as its variable approaches c
from the left and from the right.
• In other cases, the limit fails to exist because it is undefined,
such as for lim
𝒙→𝟎
𝟏
𝒙
which leads to division of 1 by zero.

31. Remark 2:
•Direction may be specified in the limit notation,
lim
𝒙→𝒄
𝒇 𝒙 by adding certain symbols.
• If x approaches c from the left, or through values less
than c, then we write lim
𝒙→𝒄−
𝒇 𝒙 .
•If x approaches c from the right, or through values
greater than c, then we write lim
𝒙→𝒄+
𝒇 𝒙 .

32. •lim
𝒙→𝒄
𝒇 𝒙 = 𝑳 if and only if
lim
𝒙→𝒄−
𝒇 𝒙 = 𝑳 and lim
𝒙→𝒄+
𝒇 𝒙 = 𝑳
•Therefore, lim
𝒙→𝒄
𝒇 𝒙 𝑫𝑵𝑬 whenever
lim
𝒙→𝒄−
𝒇 𝒙 ≠ lim
𝒙→𝒄+
𝒇 𝒙
•These limits lim
𝒙→𝒄−
𝒇 𝒙 and lim
𝒙→𝒄+
𝒇 𝒙 , are also
referred to as one-sided limits, since you only
consider values on one side of c.

33. Activity