This document defines limits in mathematics and discusses different types of limits. It begins by explaining that a limit describes what a function value approaches as the input gets arbitrarily close to a given number. The document then provides a formal definition and examples of limits, including one-sided limits. It discusses how to evaluate limits of algebraic functions, infinite limits, and limits of polynomial division. Examples are provided throughout to illustrate limit calculations.

1. 2012/2013 M. Ilyas Hadikusuma,
M.Eng
Matematika Terapan 1
TIF 21101
APPLIED MATH 1
(MATEMATIKA TERAPAN 1)
Week 11
Limits

2. 2012/2013 M. Ilyas Hadikusuma,
M.Eng
Matematika Terapan 1
Limits
Overview
Sometimes we meet the words “almost” or
“approach” in our daily life. For examples :
“Today, I almost late for the class”
or
“He almost got an accident when he tried to pass
that fast motorbike”, etc.
In math, the word almost or approach is known in
term of Limits.
Do you know what the term of limits for?

3. 2012/2013 M. Ilyas Hadikusuma,
M.Eng
Matematika Terapan 1
Limits
Definition
.)(0
)(lim
εδ
δ
ε
<−<−<
>
>=
→
Lxfcx
Lxf
cx
then,
ifthatso0aexiststhere
0eachforthatmeansstatementThe
number.realabeLletandc)atpossibly(except
ccontainingintervalopenanondefinedfunctionabefLet
c
δ+cδ−c
( )
L
ε+L
ε−L
()

4. 2012/2013 M. Ilyas Hadikusuma,
M.Eng
Matematika Terapan 1
Limits
If f is a function, then we say:
A is the limit of f (x) as x approaches c if the value of f (x)
gets arbitrarily close to A as x approaches c.
This is written in mathamatical notation as:
Notice the important fact that whether or not the notation
above is true, does not depend upon the value of f (x) when
x = c. In fact, f (x) need not even be defined when x = c. In
other word, there’s no limit if the limit of function equals to
0.
c

5. 2012/2013 M. Ilyas Hadikusuma,
M.Eng
Matematika Terapan 1
Limits
For example:
However, using factorization, we can
simplify the function into :

6. 2012/2013 M. Ilyas Hadikusuma,
M.Eng
Matematika Terapan 1
Limits
Limit of a function can be identified if the
value of left limit and right limit is equal.
Ex. :
A function is given by :
Check if the function is exist
4x, if x ≤ 4
4x + 6, if x > 4

7. 2012/2013 M. Ilyas Hadikusuma,
M.Eng
Matematika Terapan 1
Limits

8. 2012/2013 M. Ilyas Hadikusuma,
M.Eng
Matematika Terapan 1
Limits
Try this,
Using table :
From the table, the right and left limit show
the nearest value to 6, therefor there is a
limit value for x3

9. 2012/2013 M. Ilyas Hadikusuma,
M.Eng
Matematika Terapan 1
Limits
Algebraic Function of Limit

10. 2012/2013 M. Ilyas Hadikusuma,
M.Eng
Matematika Terapan 1
Limits

11. 2012/2013 M. Ilyas Hadikusuma,
M.Eng
Matematika Terapan 1
Limits
Exercises:

12. 2012/2013 M. Ilyas Hadikusuma,
M.Eng
Matematika Terapan 1
Limits
Infinite Function of Limit
It will result in : ∞ - ∞ or ∞/ ∞
Rules :

13. 2012/2013 M. Ilyas Hadikusuma,
M.Eng
Matematika Terapan 1
Limits
For division of polynomial, there are some
rules :

14. 2012/2013 M. Ilyas Hadikusuma,
M.Eng
Matematika Terapan 1
Limits
Ex. :
=

15. 2012/2013 M. Ilyas Hadikusuma,
M.Eng
Matematika Terapan 1
Limits
Exercises :