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

TIF 21101APPLIED MATH 1

(MATEMATIKA TERAPAN 1)

Week 11Limits

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

Limits

OverviewSometimes 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?

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

Limits

Definition

.)(0

)(lim

Lxfcx

Lxfcx

then ,

ifthat so 0 a exists there

0 eachfor that means statement The

number. real a beL let and c)at possibly (except

c containing interval open an on defined function a be fLet

c

cc

( )

L

L

L

( )

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

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

Limits

For example:

However, using factorization, we can simplify the function into :

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

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

Limits

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 x3

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

Limits

Algebraic Function of Limit

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

Limits

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

Limits

Exercises:

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

Limits

Infinite Function of Limit

It will result in : - or / Rules :

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

Limits

For division of polynomial, there are some rules :

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

LimitsEx. :

=

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

Limits

Exercises :