limits notes student version

8/19/2019 Limits Notes Student Version

1/36

Limit of a Function


2/36

Limit: Definition

and

Properties


3/36

It is recommended to watch this video which

going to start after a few seconds. Otherwise

please go to the next slide.

Some of the denitions refer to thereference book “Calculus – J Stwart _7th Edition”.

Introduction to imits !"#$_m%e&'_(().a*i


4/36

Definition: Suppose f(x) is defined when x is near the

number . (This means that f is defined on some open

interval that contains a, except possibly at a itself.

Then we write

lim ( x a

f x L→

=

and say !the limit of f(x), as x approaches a, e"ual L#

if we can ma$e the values of f(x) arbitrarily close to L

(as close to L as we li$e by ta$ing x to be sufficiently

close to a (on either side of a but not e"ual to a.


5/36

%otice the phrase !but x≠a” in the definition of limit. This

means that in finding the limit of f(x) as x approaches a, we

never consider x=a. In fact, f(x) needs not even be defined

when x=a. The only thing that matters is how f is defined

near a.

&ollowing figures show the graph of three functions. %ote

that in part (c, f(a) is not defined and in part (b), f(a)≠L.

'ut in each case, regardless of what happens at a, it is true

that lim ( x a

f x L→

=


6/36

Infinite Limits

et f be a function defined on both side of a, except

possibly at a itself. Then

means that the value of f(x) can be made arbitrarily

large (as large as we please by ta$ing x sufficiently

close to a, but not e"ual to a.

The symbol ) is not a number, but the expression is

often read as

!the limit of f(x), as x approaches a, is infinity

(positive or negative#

lim ( or x a

f x→

= ∞ − ∞


7/36

Infinite Limits







often read as



lim ( or x a f x→ = ∞ − ∞


8/36

Infinite Limits







often read as



lim ( or x a

f x→

= ∞ − ∞


9/36

Infinite Limits

et f be a function defined on some interval (a,) or

(*),a. Then

means that the value of f(x) can be made arbitrarilyclose to L by ta$ing x sufficiently large or large

negative.

lim ( x

f x L→±∞

=


10/36

'asically to evaluate a limit of a function we use the

following theorem.

Direct Substitution Property:

lim ( ( x a

f x f a→

=

'ut sometimes we get some $ind the answers which

are not clear such as (they may have different

values in different functions. They called

indeterminate limits.

To know how we find the answer for indeterminate

limit please go to through the next section.

+,

+

∞∞


11/36

Indeterminate Limit

Part 1:+

+


12/36

To evaluate ,

first substitute x by a so .

If , then

( lim

( x a

f x

g x→

( ( lim( ( x a

f x f a g x g a→

=

( ( + f a g a= =

( ( +lim

( ( + x a

f x f a

g x g a→= =

can be stated verbally as a very small value

divided by a very small value. Therefore it

may have different values based on variety

functions.

nl! two t!pes of them are e"aluated in this section.

+

+


13/36

Type 1: f(x) and g(x) are polynomials.

- - +(

n nn n f x a x a x a x a x a−−= + + + + ++

, -

, - , +( m m

m m g x b x b x b x b x b

−−= + + + + ++

hen , so

f(a)=+ and g(a)=+ . hich means f(x) and g(x)

have a factor of (x#a) based on the remainder

theorem.

( ( +lim

( ( + x a

f x f a

g x g a→= =


14/36


/ccording to the division theorem:

,( ( ( f x x a $ x= −

-( ( ( g x x a $ x= −

and can be found from long division. ,( $ x - ( $ x

- - -

( ( ( ( ( lim lim lim

( ( ( ( ( x a x a x a

x a $ x $ x $ a f x

g x x a $ x $ x $ a→ → →

−= = =

−


15/36


0xample: 0valuate the following limit.1 2 -

-

- lim

x

x x x x

x→−

− − + −−

STEP 1, x must be substituted by *.

( ) ( ) ( ) ( )

( )

1 2 -1 2 -

--,

- - +lim

+ x

x x x x

x→−

− − − − − + − −− − + −= =

− − −

So this limit is indeterminate limit, since it is and

numerator and denominator are polynomials, factortheorem must be applied to factori3e both and find a

factor of (x*(* or (x4.

+

+


16/36

STEP 2, factori3e the numerator and denominator:

Long division is used to factori3e .

( )

( )

( )

2 -

1 2 -

1 2

2 -

2 -

-

-

- -

- -

+

x x

x x x x x

x x

x x x

x x

x x

− +

+ − − + −− +

− − + −

− − −

−− −

( ) ( )1 2 - 2 -- - x x x x x x x− − + − = + − +

So ,

%ote that it may use identities to factori3e also (li$e denominator:


17/36

( ) ( )( ) ( )

2 -1 2 -

-

- - lim lim

x x

x x x x x x x

x x x→− →−

+ − +− − + − =− + −

STEP , rewrite the limit by using the factori3ed form:

STEP !, simplify the limit:

( )( )

2 -1 2 -

-

- - lim lim x x

x x x x x x x x→− →−

− +− − + − =− −

STEP ", substitute x by *:

%ote: after the last step if the answer

is again so must go bac$ to step -

and follow the steps again.

+

+


18/36

Type 2: f(x) or g(x) include s#uare root function.

( )

-

-

50S

. ( example: 2 , - - 6

-. ( ( : 7

2. ( ( : 6 2 ,

%O

. ( : 1

f x a x x

f x g x example x x

f x g x example x x x

f x example x

− − + −

+ − + −

− + − − − −

−


19/36


To evaluate this type of the limits (finding a factor of

(x#a, it must be multiplied by con$ugate of the

s"uare root function.

This approach refers to the following identity:

( ) ( ) ( )- -

a b a b a b− + = −&or example,

( ) ( )

( ) ( )( )( ) ( )( )

- -

- - 2 - - 2

- - 2

- - 2

6

x x x x

x x

x x

x

+ − − + + −

= + − −

= + − −

= − +

%otice that there is no s&uare root

after multiplied b! con'ugate.


20/36


0xample: 0valuate the following limit.

1

-lim

6 1 7→−

+ − − x x

x x

STEP 1, x must be substituted by 1.

So this limit is indeterminate limit, since it is so a

factor of (x#) should be found for numerator and

denominator. Since both of them include s"uare roots

then it must be multiplied by con8ugate of both.

1

- - 1 +lim

+6 1 7 1 6 1 1 7 x

x

x x→− −= =

+ − − + − × −+

+


21/36

STEP 2, multiply the numerator by its con8ugate also the

denominator by its own con8ugate.

%otice t&at since t&e #uestion must remain t&e same so'&en it is multiplied by its con$ugate t&en must be divided.

1

- - 6 1 7lim

6 1 7 - 6 1 7 x

x x x x

x x x x x→

− + + − −× ×

+ − − + + − −

( ) ( )( )( ) ( )

( ) ( )( ) ( )1 1

1 6 1 7 1 6 1 7

lim lim6 1 7 - 2 - - x x

x x x x x x

x x x x x→ →

− + − − − + − −

=+ − − + − + +

(nly t&e t'o con$ugate factors e)pand and t&e ot&er

factors remain as multiplications. *do not e)pand any+


22/36

STEP , rewrite the limit by using the factori3ed form:

( ) ( )( ) ( )1 1

1 6 1 7-lim lim6 1 7 2 1 - x x

x x x x x x x x→ →

− + + −− =+ − − − + +

STEP !, simplify the limit:

( )( )1 16 1 7-lim lim

6 1 7 2 - x x x x x

x x x→ →+ + −− =

+ − − +

STEP ", substitute x by 1:

( )( )1 16 1 7

- 9 lim lim2 1 -6 1 7 2 - x x

x x

x x x x→ →

+ + −− = = =×+ − − +


23/36

Indeterminate Limit

Part 2:∞

∞


24/36

In , as x becomes large, both numerator

and denominator become large, so it is not obvious

what happens to their ratio.

To evaluate the limit at infinity of any rational

function, we first factori,e t&e &ig&est po'er of x

occurs in t&e numerator also t&e &ig&est po'er of x

occurs in t&e denominator.

The main reason to factori3e is this theorem :

If r + is a rational number, then

If r + is a rational number such that is defined for

all x, then

( lim

( x

f x

g x→±∞

lim +

r x x→∞

=

lim +

r x x→−∞=

r x


25/36

-

- +( n n

n n f x a x a x a x a x a−

−= + + + + ++

, -

, - , +( m m

m m g x b x b x b x b x b

−−= + + + + ++

- - +

-

- +

( lim lim(

n nn n

m m x xm m

a x a x a x a x a f x g x b x b x b x b x b

−−−→±∞ →±∞

−

+ + + + +=+ + + + +

+ +

Then

hen x becomes larger both functions become

larger and it could be shown as and called

indeterminate limit. (%ote that the we may have

positive or negative infinity

∞

∞



26/36

To evaluate, we factori3e the highest power of x

from numerator and denominator.

+-

-

+-

-

( lim lim

(

n nn n n n

x xm m

m m m m

a aa a x a

f x x x x x

b bb b g x

x b x x x x

−− −

→±∞ →±∞−

− −

+ + + + + ÷ =

+ + + + + ÷

+

+

'ased on the theorem all the terms except the first one

approach to 3ero.

+-

-

+-

-

( lim lim(

n nn n n n

x xm m

m m m m

a aa a x a

f x x x x xb bb b g x

x b x x x x

−− −

→±∞ →±∞−

− −

+ + + + + ÷ = + + + + + ÷

+

+

(

(



27/36

-

- +

-

- +

lim limn n n

n n n

m m m x xm m m

a x a x a x a x a a x

b x b x b x b x b b x

−−

−→±∞ →±∞−

+ + + + +=

+ + + + ++

+

So the limit should be simplified as:

So the limit is evaluated based on the three following cases:

lim

+

n

n n

m x

m m

or if n m

a x aif n m

b x bif n m

→±∞

∞ − ∞ >= =


28/36

0xample: 0valuate the following limits.

- -- -. lim lim lim -

- x x x

x x x

x x→−∞ →−∞ →−∞

−= = = −∞

+

- -

- -- - --. lim lim2 - 2 2 x x x x x x→−∞ →−∞

− = =+

- -

2 2

- - -2. lim lim lim +

- x x x

x x

x x x→−∞ →−∞ →−∞

−= = =

+



29/36


STEP 1, factori3e the highest power of x inside the

root function.

%ote that :

;an be written as:

, -

, - , +lim n n

n n x

a x a x a x a x a−

−→±∞+ + + + ++

, +- ,

- ,lim

n nn n n n x

a aa a x a

x x x x

−− −→±∞

+ + + + + ÷

+

so

, -

, - , +lim limn n n

n n n x x

a x a x a x a x a a x−−

→±∞ →±∞

+ + + + + =+


30/36


Important noteWe know that square root of any number is a

positive number (it is impossible to be

negative) which means that if the sign of thevariable is not clear we have to use absolute

function.

In this part we need to know

( )-

( ( f x f x=

- x x=


31/36


Thereforehen then is substituted by x.

hen then is substituted by * x.

x → ∞ - x

x → −∞ - x

STEP 2, factori3e the highest power of x for numerator and

denominator and simplify.

%otice t&at multiplying by con$ugate is not applicable for

t&is type of indeterminate limit.


32/36

0xample: 0valuate the limit.


-1 lim

- 2 x

x x

x→∞+ +

−

&irst, factori3e the highest power of x for root function

--- 11

lim lim- 2 - 2 x x

x x x x x

x x→∞ →∞

+ + ÷+ + =− −

Then simplify the root function

-

- -

1-1

lim lim lim- 2 - 2 - 2 x x x

x x x x x x x

x x x→∞ →∞ →∞

+ + ÷ ++ = = =− − −

(


33/36


Since then x → ∞

- - 2lim lim lim

- 2 - 2 - 2 x x x

x x x x x

x x x→∞ →∞ →∞

+ += = =

− − −

%ow , we factori3e the highest power of x from both as

usual.

2 2 2 2lim lim lim

2- 2 - --

x x x

x x x

x x x

x

→∞ →∞ →∞= = = =

− − ÷

What if x → −∞


34/36

0xample: 0valuate the limit.


-1 lim

- 2 x x x

x→−∞+ +

−

&irst, factori3e the highest power of x for root function

-

--

11

lim lim- 2 - 2 x x

x x x x x

x x→−∞ →−∞

+ + ÷+ + =− −

Then simplify the root function

-

- -1

-1lim lim lim

- 2 - 2 - 2 x x x

x x x x x x x

x x x→−∞ →−∞ →−∞

+ + ÷ ++ = = =− − −

(


35/36


Since then x → −∞

- -lim lim lim

- 2 - 2 - 2 x x x

x x x x x

x x x→−∞ →−∞ →−∞

+ − −= = =

− − −

%ow , we factori3e the highest power of x from both as

usual.

,lim lim lim

2- 2 - --

x x x

x x x

x x x

x

→−∞ →−∞ →−∞

− − − −= = = =

− − ÷


36/36

Thank you,

I hope this notes can help students tounderstand more about limit of a functions.

Please do not hesitate to email me( [email protected] ) your useful comments

and feedbacks.

Prepared by: Mohammad eza !eh"hani #enior $ecturer %entre for &oundation#tudies 'niersity Tunku bdulahman ('T)

limits notes student version

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