M53 Lec1.1 Limits-OneSided1

8/12/2019 M53 Lec1.1 Limits-OneSided1

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Limit of a Function and

One-sided limits

Mathematics 53

Institute of Mathematics (UP Diliman)

Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 1 / 39


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For today

1 Limit of a Function: An intuitive approach

2 Evaluating Limits

3 One-sided Limits



3/289

For today


2 Evaluating Limits

3 One-sided Limits



4/289

Introduction

Given a function f(x)anda

,



5/289

Introduction


,

what is the value of f atxneara,



6/289

Introduction


,

what is the value of f atxneara,

but not equal toa?



7/289

Illustration 1

Consider f(x) =3x 1.



8/289

Illustration 1

Consider f(x) =3x 1.What can we say about values of f(x)for values ofxnear1 but not equal to 1?



9/289

Illustration 1


x f(x)



10/289

Illustration 1


x f(x)

0



11/289

Illustration 1


x f(x)

0 1



12/289

Illustration 1


x f(x)

0 10.5



13/289

Illustration 1


x f(x)

0 10.5 0.5



14/289

Illustration 1


x f(x)

0 10.5 0.5

0.9



15/289

Illustration 1


x f(x)

0 10.5 0.5

0.9 1.7



16/289

Illustration 1


x f(x)

0 10.5 0.5

0.9 1.7

0.99 1.97

0.99999 1.99997



17/289

Illustration 1


x f(x)

0 10.5 0.5

0.9 1.7

0.99 1.97

0.99999 1.99997

x f(x)



18/289

Illustration 1


x f(x)

0 10.5 0.5

0.9 1.7

0.99 1.97

0.99999 1.99997

x f(x)

2



19/289

Illustration 1


x f(x)

0 10.5 0.5

0.9 1.7

0.99 1.97

0.99999 1.99997

x f(x)

2 5



20/289

Illustration 1


x f(x)

0 10.5 0.5

0.9 1.7

0.99 1.97

0.99999 1.99997

x f(x)

2 51.5 3.5

1.1 2.3

1.001 2.003

1.00001 2.00003



21/289

Illustration 1


x f(x)

0 10.5 0.5

0.9 1.7

0.99 1.97

0.99999 1.99997

x f(x)

2 51.5 3.5

1.1 2.3

1.001 2.003

1.00001 2.00003

Based on the table, as x gets closer and closer to 1,



22/289

Illustration 1


x f(x)

0 10.5 0.5

0.9 1.7

0.99 1.97

0.99999 1.99997

x f(x)

2 51.5 3.5

1.1 2.3

1.001 2.003

1.00001 2.00003

Based on the table, as x gets closer and closer to 1, f(x)gets closer and closer

to2.


Ill i 1


23/289

Illustration 1

x f(x)

0 10.5 0.5

0.9 1.7

0.99 1.97

0.99999 1.99997

1 1 2 3

1

1

2

3

4



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Ill t ti 1


25/289

Illustration 1

x f(x)

0 10.5 0.5

0.9 1.7

0.99 1.97

0.99999 1.99997

1 1 2 3

1

1

2

3

4


Ill stration 1


26/289

Illustration 1

x f(x)

0 10.5 0.5

0.9 1.7

0.99 1.97

0.99999 1.99997

1 1 2 3

1

1

2

3

4


Illustration 1


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Illustration 1

x f(x)

0 10.5 0.5

0.9 1.7

0.99 1.97

0.99999 1.99997

1 1 2 3

1

1

2

3

4


Illustration 1


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Illustration 1

x f(x)

2 5

1.5 3.5

1.1 2.3

1.001 2.003

1.00001 2.00003

1 1 2 3

1

1

2

3

4


Illustration 1


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Illustration 1

x f(x)

2 5

1.5 3.5

1.1 2.3

1.001 2.003

1.00001 2.00003

1 1 2 3

1

1

2

3

4


Illustration 1


30/289

Illustration 1

x f(x)

2 5

1.5 3.5

1.1 2.3

1.001 2.003

1.00001 2.00003

1 1 2 3

1

1

2

3

4


Illustration 1


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Illustration 1

x f(x)

2 5

1.5 3.5

1.1 2.3

1.001 2.003

1.00001 2.00003

1 1 2 3

1

1

2

3

4


Illustration 1


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Illustration 1

x f(x)

2 5

1.5 3.5

1.1 2.3

1.001 2.003

1.00001 2.00003

1 1 2 3

1

1

2

3

4


Illustration 1


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Illustration 1

x f(x)

2 5

1.5 3.5

1.1 2.3

1.001 2.003

1.00001 2.00003

1 1 2 3

1

1

2

3

4

Asxgets closer and closer to 1, f(x)gets closer and closer to 2.


Illustration 2


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Illustration 2

Consider: g(x) = 3x2 4x+1

x 1


Illustration 2


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Illustration 2


x 1=

(3x 1)(x 1)x 1


Illustration 2


36/289

Illustration 2


x 1=

(3x 1)(x 1)x 1

=3x 1, x = 1


Illustration 2


37/289


x 1=

(3x 1)(x 1)x 1

=3x 1, x = 1

1 1 2 31

1

2

3

4


Illustration 2


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x 1=

(3x 1)(x 1)x 1

=3x 1, x = 1

1 1 2 31

1

2

3

4

Asxgets closer and closer to 1,


Illustration 2


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x 1=

(3x 1)(x 1)x 1

=3x 1, x = 1

1 1 2 31

1

2

3

4



Illustration 2


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x 1=

(3x 1)(x 1)x 1

=3x 1, x = 1

1 1 2 31

1

2

3

4



Illustration 2


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x 1=

(3x 1)(x 1)x 1

=3x 1, x = 1

1 1 2 31

1

2

3

4



Illustration 2


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x 1=

(3x 1)(x 1)x 1

=3x 1, x = 1

1 1 2 31

1

2

3

4

Asxgets closer and closer to 1, g(x)gets closer and closer to 2.


Illustration 3


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Consider: h(x) =

3x 1, x =10, x=1


Illustration 3


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Consider: h(x) =

3x 1, x =10, x=1

1 1 2 31

1

2

3

4


Illustration 3


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Consider: h(x) =

3x 1, x =10, x=1

1 1 2 31

1

2

3

4



Illustration 3


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Consider: h(x) =

3x 1, x =10, x=1

1 1 2 31

1

2

3

4



Illustration 3


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Consider: h(x) =

3x 1, x =10, x=1

1 1 2 31

1

2

3

4



Illustration 3


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Consider: h(x) =

3x 1, x =10, x=1

1 1 2 31

1

2

3

4



Illustration 3


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Consider: h(x) =

3x 1, x =10, x=1

1 1 2 31

1

2

3

4

Asxgets closer and closer to 1, h(x)gets closer and closer to 2.


Limit


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Limit


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Intuitive Notion of a Limit


Limit


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Intuitive Notion of a Limita ,L


Limit


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Intuitive Notion of a Limita ,L f(x): function defined on some open interval containinga, except possibly at a


Limit


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The limit of f(x)asxapproachesa isL


Limit


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The limit of f(x)

asx

approachesa

isL

if the values of f(x)get closer and closer to Lasxassumes values getting closer

and closer toabut not reachinga.


Limit


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The limit of f(x)

asx

approachesa

isL

if the values of f(x)get closer and closer to Lasxassumes values getting closer

and closer toabut not reachinga.

Notation:limxa f(x) = L


Examples


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f(x) = 3x 1

1 1 2 3

1

1

2

3

4


Examples


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f(x) = 3x 1

1 1 2 3

1

1

2

3

4 lim

x1(3x 1)


Examples


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f(x) = 3x 1

1 1 2 3

1

1

2

3

4 lim

x1(3x 1) =2


Examples


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f(x) = 3x 1

1 1 2 3

1

1

2

3

4 lim

x1(3x 1) =2

Note: In this case, limx1

f(x)


Examples


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f(x) = 3x 1

1 1 2 3

1

1

2

3

4 lim

x1(3x 1) =2

Note: In this case, limx1

f(x) = f(1).


Examples


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g(x) = 3

x

2

4

x+1

x 1

1 1 2 31

1

2

3

4


Examples


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g(x) = 3

x

2

4

x+1

x 1

1 1 2 31

1

2

3

4 limx1

3x2 4x+1x 1


Examples


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g(x) = 3x2

4x+1

x 1

1 1 2 31

1

2

3

4 limx1

3x2 4x+1x 1 =2


Examples


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g(x) = 3x2

4x+1

x 1

1 1 2 31

1

2

3

4 limx1

3x2 4x+1x 1 =2

Note: Thoughg(1)is undefined,limx1

g(x)exists.


Examples


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h(x) =

3x

1, x

=1

0, x=1

1 1 2 31

1

2

3

4


Examples


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h(x) =

3x

1, x

=1

0, x=1

1 1 2 31

1

2

3

4

limx1

h(x)


Examples


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h(x) =

3x

1, x

=1

0, x=1

1 1 2 31

1

2

3

4

limx1

h(x) =2


Examples


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h(x) =

3x

1, x

=1

0, x=1

1 1 2 31

1

2

3

4

limx1

h(x) =2

Note:h(1) = limx1 h(x).


Some Remarks


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Remark

In finding limxa f(x):


Some Remarks


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Remark


We only need to consider values of x very close toabut not exactly at a.


Some Remarks


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Remark



Thus, limxa f(x)is NOT NECESSARILYthe same as f(a).


Some Remarks


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Remark



Thus, limxa f(x)is NOT NECESSARILYthe same as f(a).

We letxapproachafrom BOTH SIDES ofa.


Some Remarks


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If f(x)

does not approach any

particular real number asx

approachesa, then we say

limxa f(x)does not exist (dne).


Some Remarks


75/289

If f(x)

does not approach any




e.g.

H(x) =

1, x 0

0, x < 0

(Heaviside Function)


Some Remarks


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If f(x)does not approach any




e.g.

H(x) =

1, x 0

0, x < 0


3 2 1 1 2 3

1

2

3

0


Some Remarks


77/289





e.g.

H(x) =

1, x 0

0, x < 0


3 2 1 1 2 3

1

2

3

0

limx0

H(x) = 0?


Some Remarks


78/289





e.g.

H(x) =

1, x 0

0, x < 0


3 2 1 1 2 3

1

2

3

0

limx0

H(x) = 0? No.


Some Remarks


79/289





e.g.

H(x) =

1, x 0

0, x < 0


3 2 1 1 2 3

1

2

3

0

limx0

H(x) = 0? No.

limx0

H(x) = 1?


Some Remarks


80/289





e.g.

H(x) =

1, x 0

0, x < 0


3 2 1 1 2 3

1

2

3

0

limx0

H(x) = 0? No.

limx0

H(x) = 1? No.


Some Remarks


81/289





e.g.

H(x) =

1, x 0

0, x < 0


3 2 1 1 2 3

1

2

3

0

limx0

H(x) = 0? No.

limx0

H(x) = 1? No.

limx0

H(x) dne


For today


82/289


2 Evaluating Limits

3 One-sided Limits


Limit Theorems


83/289

Theorem


Limit Theorems


84/289

Theorem

If limxa f(x)exists, then it is unique.


Limit Theorems


85/289

Theorem


Ifc , then limxa c


Limit Theorems


86/289

Theorem


Ifc , then limxa c=c.


Limit Theorems


87/289

Theorem


Ifc , then limxa c=c.limxa x


Limit Theorems


88/289

Theorem


Ifc , then limxa c=c.limxa x=a


Limit Theorems


89/289

Theorem

Suppose limxa f(x) =L1 and limxag(x) = L2. Letc ,n .


Limit Theorems


90/289

Theorem


limxa[f(x)g(x)]


Limit Theorems


91/289

Theorem


limxa[f(x)g(x)] = limxa f(x)


Limit Theorems


92/289

Theorem


limxa[f(x)g(x)] = limxa f(x) limxag(x)


Limit Theorems


93/289

Theorem


limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2


Limit Theorems


94/289

Theorem


limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2limxa

[f(x)g(x)]


Limit Theorems


95/289

Theorem



[f(x)g(x)] = limxa

f(x) limxa

g(x)


Limit Theorems


96/289

Theorem



[f(x)g(x)] = limxa

f(x) limxa

g(x) =L1 L2

Institute of Mathematics (UP Diliman) Limit of a Function and One sided limits Mathematics 53 18 / 39

Limit Theorems

Th


97/289

Theorem



[f(x)g(x)] = limxa

f(x) limxa

g(x) =L1 L2limxa[c f(x)] =


Limit Theorems

Th


98/289

Theorem



[f(x)g(x)] = limxa

f(x) limxa

g(x) =L1 L2limxa[c f(x)] =climxa f(x)


Limit Theorems

Th


99/289

Theorem



[f(x)g(x)] = limxa

f(x) limxa

g(x) =L1 L2limxa[c f(x)] =climxa f(x) =cL1


Limit Theorems

Theorem


100/289

Theorem


limxa[f(x)g(x)] = limxa f(x) limxag(x) = L1 L2lim

xa[f(x)g(x)] = lim

xaf(x) lim

xag(x) =L1 L2

limxa[c f(x)] =climxa f(x) =cL1

limxa

f(x)

g(x)


Limit Theorems

Theorem


101/289

Theorem



xa[f(x)g(x)] = lim

xaf(x) lim

xag(x) =L1 L2


limxa

f(x)

g(x) =

limxa f(x)

limxag(x)


Limit Theorems

Theorem


102/289

Theorem



xa[f(x)g(x)] = lim

xaf(x) lim

xag(x) =L1 L2


limxa

f(x)

g(x) =

limxa f(x)

limxag(x)

= L1L2


Limit Theorems

Theorem


103/289

Theorem



xa[f(x)g(x)] = lim

xaf(x) lim

xag(x) =L1 L2


limxa

f(x)

g(x) =

limxa f(x)

limxag(x)

= L1L2

, providedL2=0


Limit Theorems

Theorem


104/289

Theorem



xa[f(x)g(x)] = lim

xaf(x) lim

xag(x) =L1 L2


limxa

f(x)

g(x) =

limxa f(x)

limxag(x)

= L1L2

, providedL2=0

limxa (f(x))

n


Limit Theorems

Theorem


105/289

Theorem



xa[f(x)g(x)] = lim

xaf(x) lim

xag(x) =L1 L2


limxa

f(x)

g(x) =

limxa f(x)

limxag(x)

= L1L2

, providedL2=0

limxa (f(x))

n =

limxa f(x)

n


Limit Theorems

Theorem


106/289

Theorem



xa[f(x)g(x)] = lim

xaf(x) lim

xag(x) =L1 L2


limxa

f(x)

g(x) =

limxa f(x)

limxag(x)

= L1L2

, providedL2=0

limxa (f(x))

n =

limxa f(x)

n = (L1)n


Evaluate: limx1

(2x2 +3x 4)


107/289


Evaluate: limx1

(2x2 +3x 4)

limx1

(2x2 +3x 4)


108/289



109/289

Evaluate: limx1

(2x2 +3x 4)

limx1

(2x2 +3x 4) = limx1

2x2 +


110/289


Evaluate: limx1

(2x2 +3x 4)

limx1

(2x2 +3x 4) = limx1

2x2 + limx1

3x


111/289


Evaluate: limx1

(2x2 +3x 4)

limx1

(2x2 +3x 4) = limx1

2x2 + limx1

3x


112/289


Evaluate: limx1

(2x2 +3x 4)

limx1

(2x2 +3x 4) = limx1

2x2 + limx1

3x limx1

4


113/289


Evaluate: limx1

(2x2 +3x 4)

limx1

(2x2 +3x 4) = limx1

2x2 + limx1

3x limx1

4


114/289

=2

limx1 x2


Evaluate: limx1

(2x2 +3x 4)

limx1

(2x2 +3x 4) = limx1

2x2 + limx1

3x limx1

4


115/289

=2

limx1 x2

+3

limx1 x


Evaluate: limx1

(2x2 +3x 4)

limx1

(2x2 +3x 4) = limx1

2x2 + limx1

3x limx1

4


116/289

=2

limx1 x2

+3

limx1 x limx1 4


Evaluate: limx1

(2x2 +3x 4)

limx1

(2x2 +3x 4) = limx1

2x2 + limx1

3x limx1

4

2


117/289

=2

limx1 x2

+3

limx1 x limx1 4

=2

limx1

x

2


Evaluate: limx1

(2x2 +3x 4)

limx1

(2x2 +3x 4) = limx1

2x2 + limx1

3x limx1

4

2


118/289

=2

limx1 x2

+3

limx1 x limx1 4

=2

limx1

x

2+3

limx1

x

lim

x14


Evaluate: limx1

(2x2 +3x 4)

limx1

(2x2 +3x 4) = limx1

2x2 + limx1

3x limx1

4

2


119/289

=2

limx1 x2

+3

limx1 x limx1 4

=2

limx1

x

2+3

limx1

x

lim

x14

=2


Evaluate: limx1

(2x2 +3x 4)

limx1

(2x2 +3x 4) = limx1

2x2 + limx1

3x limx1

4

2


120/289

=2

limx1 x

+3

limx1 x limx1 4

=2

limx1

x

2+3

limx1

x

lim

x14

=2(

1)2


Evaluate: limx1

(2x2 +3x 4)

limx1

(2x2 +3x 4) = limx1

2x2 + limx1

3x limx1

4

2


121/289

=2

limx1 x

+3

limx1 x limx1 4

=2

limx1

x

2+3

limx1

x

lim

x14

=2(

1)2 +3


Evaluate: limx1

(2x2 +3x 4)

limx1

(2x2 +3x 4) = limx1

2x2 + limx1

3x limx1

4

2


122/289

=2

limx1 x

+3

limx1 x limx1 4

=2

limx1

x

2+3

limx1

x

lim

x14

=2(

1)2 +3(

1)


Evaluate: limx1

(2x2 +3x 4)

limx1

(2x2 +3x 4) = limx1

2x2 + limx1

3x limx1

4

l2

l l


123/289

=2

limx1 x

+3

limx1 x limx1 4

=2

limx1

x

2+3

limx1

x

lim

x14

=2(

1)2 +3(

1)

4



124/289

Evaluate: limx1

(2x2 +3x 4)

limx1

(2x2 +3x 4) = limx1

2x2 + limx1

3x limx1

4

2

li2

3

li

li 4


125/289

=2

limx1 x

+3

limx1 x limx1 4

=2

limx1

x

2+3

limx1

x

lim

x14

=2(

1)2 +3(

1)

4

= 5

In general:

Remark

If f is a polynomial function, then limxa f(x)


Evaluate: limx1

(2x2 +3x 4)

limx1

(2x2 +3x 4) = limx1

2x2 + limx1

3x limx1

4

2

li2

3

li

li 4


126/289

=2

limx1 x

+3

limx1 x limx1 4

=2

limx1

x

2+3

limx1

x

lim

x14

=2(

1)2 +3(

1)

4

= 5

In general:

Remark

If f is a polynomial function, then limxa f(x) = f(a).


Evaluate: limx2

4x3 +3x2 x+1x2 +2


127/289


Evaluate: limx2

4x3 +3x2 x+1x2 +2

li 4x3 + 3x2 x + 1


128/289

limx2

4x3 +3x2 x+1x2 +2


Evaluate: limx2

4x3 +3x2 x+1x2 +2

li 4x3

+ 3x2

x + 1lim

x 2(4x3 +3x2 x+1)


129/289

limx2

4x +3x x+1x2 +2

= x2 lim

x2(x2 +2)


Evaluate: limx2

4x3 +3x2 x+1x2 +2

li 4x3

+ 3x2

x + 1lim

x2(4x3 +3x2 x+1)


130/289

limx2

4x +3x x+1x2 +2

= x2 lim

x2(x2 +2)

= 4(8)


Evaluate: limx2

4x3 +3x2 x+1x2 +2

lim 4x3

+ 3x2

x + 1lim

x2(4x3 +3x2 x+1)


131/289

limx2

4x +3x x+1x2 +2

= x 2 lim

x2(x2 +2)

= 4(8) +3(4)


Evaluate: limx2

4x3 +3x2 x+1x2 +2

lim 4x3

+ 3x2

x + 1 =lim

x2(4x3 +3x2

x+1)


132/289

limx2

4x +3x x+1x2 +2

= x 2 lim

x2(x2 +2)

= 4(8) +3(4) (2) +1


Evaluate: limx2

4x3 +3x2 x+1x2 +2

lim 4x3

+ 3x2

x + 1 =lim

x2(4x3 +3x2

x+1)


133/289

limx2

4x +3x x+1x2 +2

= x 2lim

x2(x2 +2)

= 4(8) +3(4) (2) +1

4+2


Evaluate: limx2

4x3 +3x2 x+1x2 +2

lim 4x3

+ 3x2

x + 1 =lim

x2(4x3 +3x2

x+1)


134/289

limx2

4x +3x x+1x2 +2

=lim

x2(x2 +2)

= 4(8) +3(4) (2) +1

4+2

= 176


Evaluate: limx2

4x3 +3x2 x+1x2 +2

lim 4x3

+ 3x2

x + 1 =lim

x2(4x3 +3x2

x+1)


135/289

limx2

4x +3x x+1x2 +2

=lim

x2(x2 +2)

= 4(8) +3(4) (2) +1

4+2

= 176

Remark

If fis a rational function and f(a)is defined,


Evaluate: limx2

4x3 +3x2 x+1x2 +2

lim 4x3

+ 3x2

x + 12

=lim

x2(4x3 +3x2

x+1)

2


136/289

limx2

4x +3x x+1x2 +2

lim

x2(x2 +2)

= 4(8) +3(4) (2) +1

4+2

= 176

Remark

If fis a rational function and f(a)is defined, then lim

xaf(x) = f(a).


Theorem

Suppose limxa f(x)exists andn . Then,

limxan

f(x) = n

limxa f(x),


137/289

f ( )

f ( ),

provided limxa f(x) > 0whenn is even.


Theorem


limxan

f(x) = n

limxa f(x),


138/289

f ( )

f ( ),


limx3

3x

1


Theorem


limxa

nf(x) =

nlimxa f(x)

,


139/289

f ( )

f ( )


limx3

3x

1= limx3

(3x

1)


Theorem


limxa

nf(x) =

nlimxa f(x)

,


140/289

f ( )

f ( )


limx3

3x

1= limx3

(3x

1) =

8


Theorem


limxa

nf(x) =

nlimxa f(x)

,


141/289

f ( )

f ( )


limx3

3x

1= limx3

(3x

1) =

8= 2

2


Theorem


limxa

nf(x) =

nlimxa f(x)

,


142/289

( )

( )


limx3

3x

1= limx3

(3x

1) =

8= 2

2

limx1

3

x+4

x 2


Theorem


limxa

nf(x) =

nlimxa f(x)

,


143/289

provided lim

xa f(x) > 0whenn is even.

limx3

3x

1= lim

x3(3x

1) =

8= 2

2

limx1

3

x+4

x 2 = 3


Theorem


limxa

nf(x) =

nlimxa f(x)

,


144/289

provided lim


limx3

3x

1= lim

x3(3x

1) =

8= 2

2

limx1

3

x+4

x 2 = 3

1+41 2


Theorem


limxa

nf

(x

) = nlimxa f

(x

),


145/289

provided lim


limx3

3x

1= lim

x3(3x

1) =

8= 2

2

limx1

3

x+4

x 2 = 3

1+41 2 = 1


Theorem


limxa

nf(x) = nlimxa

f(x),


146/289

provided lim


lim

x3

3x

1= lim

x3(3x

1) =

8= 2

2

limx1

3

x+4

x 2 = 3

1+41 2 = 1

limx7/2

4

3 2x


Theorem


limxa

nf(x) = nlimxa

f(x),


147/289

provided lim


lim

x3

3x

1= lim

x3(3x

1) =

8= 2

2

limx1

3

x+4

x 2 = 3

1+41 2 = 1

limx7/2

4

3 2x dne


Theorem


limxa

nf(x) = nlimxa

f(x),


148/289

provided lim


lim

x3

3x

1= lim

x3(3x

1) =

8= 2

2

limx1

3

x+4

x 2 = 3

1+41 2 = 1

limx7/2

4

3 2x dne

limx2

x2 4


Theorem


limxa

nf(x) = nlimxa

f(x),


149/289

provided lim


lim

x3

3x

1= lim

x3(3x

1) =

8= 2

2

limx1

3

x+4

x 2 = 3

1+41 2 = 1

limx7/2

4

3 2x dne

limx2

x2 4=??


Theorem


limxa

nf(x) = nlimxa

f(x),


150/289

provided lim


lim

x3

3x

1= lim

x3(3x

1) =

8= 2

2

limx1

3

x+4

x 2 = 3

1+41 2 = 1

limx7/2

4

3 2x dne

limx2

x2 4=?? (for now)


Evaluate: limx3

2x2 5x+1

x3 x+4

3

limx32x2

5x+1

x3 x+4 3


151/289


Evaluate: limx3

2x2 5x+1

x3 x+4

3

limx32x2

5x+1

x3 x+4 3

=

limx32x2

5x+1

x3 x+4 3


152/289


Evaluate: limx3

2x2 5x+1

x3 x+4

3

limx32x2

5x+1

x3 x+4 3

=

limx32x2

5x+1

x3 x+4 3


153/289

=


Evaluate: limx3

2x2 5x+1

x3 x+4

3

limx32x2

5x+1

x3 x+4 3

=

limx32x2

5x+1

x3 x+4 3


154/289

=

limx3

2x2


Evaluate: limx3

2x2 5x+1

x3 x+4

3

limx32x2

5x+1

x3 x+4 3

=

limx32x2

5x+1

x3 x+4 3


155/289

=

limx3

2x2


Evaluate: limx3

2x2 5x+1

x3 x+4

3

limx32x2

5x+1

x3 x+4 3

=

limx32x2

5x+1

x3 x+4 3


156/289

=

limx3

2x2

limx3

(5x+1)


Evaluate: limx3

2x2 5x+1

x3 x+4

3

limx32x2

5x+1

x3 x+4 3

=

limx32x2

5x+1

x3 x+4 3


157/289

=

limx3

2x2

limx3

(5x+1)

limx

3(x3 x+4)

3


Evaluate: limx3

2x2 5x+1

x3 x+4

3

limx32x2

5x+1

x3 x+4 3

=

limx32x2

5x+1

x3 x+4 3

3


158/289

=

limx3

2x2

limx3

(5x+1)

limx

3(x3 x+4)

3

=

18


Evaluate: limx3

2x2 5x+1

x3 x+4

3

limx32x2

5x+1

x3 x+4 3

=

limx32x2

5x+1

x3 x+4 3

3


159/289

=

limx3

2x2

limx3

(5x+1)

limx

3(x3 x+4)

3

=

18 4


Evaluate: limx3

2x2 5x+1

x3 x+4

3

limx32x2

5x+1

x3 x+4 3

=

limx32x2

5x+1

x3 x+4 3

3


160/289

=

limx3

2x2

limx3

(5x+1)

limx

3(x3 x+4)

3

=

18 4

28

3


Evaluate: limx3

2x2 5x+1

x3 x+4

3

limx32x2

5x+1

x3 x+4 3

=

limx32x2

5x+1

x3 x+4 3

3


161/289

=

limx3

2x2

limx3

(5x+1)

limx

3(x3 x+4)

3

=

18 4

28

3

= 1

8



x 1 . From earlier, limx1g(x) = 2.


162/289




Can we arrive at this conclusion computationally?


163/289





Note that limx1

3x2 4x+1


164/289

x1





Note that limx1

3x2 4x+1

=0


165/289

x1





Note that limx1

3x2 4x+1

=0and lim

x1(x 1)


166/289

x1

x1





Note that limx1

3x2 4x+1

=0and lim

x1(x 1) = 0.


167/289

x1

x1





Note that limx1

3x2 4x+1

=0and lim

x1(x 1) = 0.


168/289

x x

But whenx =1, 3x2 4x+1

x

1 =

(3x 1)(x 1)x

1 =3x 1.





Note that limx1

3x2 4x+1

=0and lim

x1(x 1) = 0.


169/289


x

1 =

(3x 1)(x 1)x

1 =3x 1.

Since we are just taking the limit asx 1,



170/289




Note that limx1

3x2 4x+1

=0and lim

x1(x 1) = 0.


171/289


x

1 =

(3x 1)(x 1)x

1 =3x 1.


limx1

3x2 4x+1x 1 = limx1(3x 1)





Note that limx1

3x2 4x+1

=0and lim

x1(x 1) = 0.


172/289


x

1 =

(3x 1)(x 1)x

1 =3x 1.


limx1

3x2 4x+1x 1 = limx1(3x 1) =2.


Definition


173/289


Definition

If limxa f(x) =0and limxag(x) = 0


174/289


Definition

If limxa f(x) =0and limxag(x) = 0then

lim

xa

f(x)

g(x)

is called anindeterminate form of type 0

0.


175/289

yp0


Definition


lim

xa

f(x)

g(x)


0.


176/289

yp0

Remarks:


Definition


limx

a

f(x)

g(x)


0.


177/289

0

Remarks:

1 If f(a) = 0and g(a) =0, then f(a)g(a)

is undefined!


Definition


limx

a

f(x)

g(x)


0.


178/289

0

Remarks:


is undefined!

2 The limit above MAY or MAY NOT exist.


Definition


limx

a

f(x)

g(x)


0.


179/289

0

Remarks:


is undefined!


3 Some techniques used in evaluating such limits are:


Definition


limx

a

f(x)

g(x)


0.


180/289

0

Remarks:


is undefined!


3 Some techniques used in evaluating such limits are:

FactoringRationalization


Examples

Evaluate: lim

x1

x2 +2x+1

x+1


181/289


Examples

Evaluate: lim

x1

x2 +2x+1

x+1

0

0


182/289


Examples

Evaluate: lim

x1

x2 +2x+1

x+1

0

0

limx 1

x2 +2x+1

x + 1


183/289

x1 x+1


Examples

Evaluate: lim

x1

x2 +2x+1

x+1

0

0lim

x 1x2 +2x+1

x + 1 = lim

x 1


184/289

x1 x+1 x1



185/289

Examples

Evaluate: lim

x1

x2 +2x+1

x+1

0

0lim

x1x2 +2x+1

x + 1 = lim

x1(x+1)2

x + 1


186/289

x 1 x+1 x 1 x+1


Examples

Evaluate: lim

x1

x2 +2x+1

x+1

0

0lim

x1x2 +2x+1

x+1 = lim

x1(x+1)2

x+1


187/289

x 1 x + 1 x 1 x + 1

= limx

1(x+1)


Examples

Evaluate: limx

1

x2 +2x+1

x+1

0

0lim

x1x2 +2x+1

x+1 = lim

x1(x+1)2

x+1


188/289

+ +

= limx

1(x+1)

= (1+1)


Examples

Evaluate: limx

1

x2 +2x+1

x+1

0

0lim

x1x2 +2x+1

x+1 = lim

x1(x+1)2

x+1


189/289

= limx

1(x+1)

= (1+1)

= 0



190/289


191/289

Examples

Evaluate: limx

2

x3 +8

x2

4

0

0lim

x2x3 +8

x2 4


192/289


Examples

Evaluate: limx

2

x3 +8

x2

4

0

0lim

x2x3 +8

x2 4 = limx2


193/289


Examples

Evaluate: limx

2

x3 +8

x2

4

0

0lim

x2x3 +8

x2 4 = limx2(x+2)(x2 2x+4)


194/289


Examples

Evaluate: limx

2

x3 +8

x2

4

0

0lim

x2x3 +8

x2 4 = limx2(x+2)(x2 2x+4)

(x+2)(x 2)


195/289


Examples

Evaluate: limx

2

x3 +8

x2

4

0

0lim

x2x3 +8

x2 4 = limx2(x+2)(x2 2x+4)

(x+2)(x 2)2 2 4


196/289

= limx

2

x2 2x+4x

2


Examples

Evaluate: limx

2

x3 +8

x2

4

0

0lim

x2x3 +8

x2 4 = limx2(x+2)(x2 2x+4)

(x+2)(x 2)2 2 4


197/289

= limx

2

x2 2x+4x

2

= 4+4+4

2 2


Examples

Evaluate: limx

2

x3 +8

x2

4

0

0lim

x2x3 +8

x2 4 = limx2(x+2)(x2 2x+4)

(x+2)(x 2)2 2 + 4


198/289

= limx

2

x2 2x+4x

2

= 4+4+4

2 2= 3


Examples

Evaluate: limx4

x2 162x


199/289


Examples

Evaluate: limx4

x2 162x

0

0


200/289


Examples

Evaluate: limx4

x2 162x

0

0

limx4x2

16

2x


201/289


Examples

Evaluate: limx4

x2 162x

0

0

limx4x2

16

2x = limx4x2

16

2x


202/289


Examples

Evaluate: limx4

x2 162x

0

0

limx4x2

16

2x = limx4x2

16

2x 2+

x

2+x


203/289


Examples

Evaluate: limx4

x2 162x

0

0

limx4x2

16

2x = limx4x2

16

2x 2+

x

2+x

= limx4

(x2 16)(2+x)


204/289


Examples

Evaluate: limx4

x2 162x

0

0

limx4x2

16

2x = limx4x2

16

2x 2+

x

2+x

= limx4

(x2 16)(2+x)4 x


205/289



206/289

Examples

Evaluate: limx4

x2 162x

0

0

limx4x2

16

2x = limx4x2

16

2x 2+

x

2+x

= limx4

(x2 16)(2+x)4 x


207/289

= limx4

(x 4)(x+4)(2+x)4 x

= limx4

[(x+4)(2+x)]


Examples

Evaluate: limx4

x2 162x

0

0

limx4x2

16

2x = limx4x2

16

2x 2+

x

2+x

= limx4

(x2 16)(2+x)4 x

( )( )(

)


208/289

= limx4

(x 4)(x+4)(2+x)4 x

= limx4

[(x+4)(2+x)]

= (8)(4)


Examples

Evaluate: limx4

x2 162x

0

0

limx4x2

16

2x = limx4x2

16

2x 2+

x

2+x

= limx4

(x2 16)(2+x)4 x

( )( )(

)


209/289

= limx4

(x 4)(x+4)(2+x)4 x

= limx4

[(x+4)(2+x)]

= (8)(4)

= 32


Examples

Evaluate: limx8

3

x 2x2 7x 8


210/289


Examples

Evaluate: limx8

3

x 2x2 7x 8

0

0


211/289


Examples

Evaluate: limx8

3

x 2x2 7x 8

0

0

limx8

3

x 2x2 7x 8


212/289


Examples

Evaluate: limx8

3

x 2x2 7x 8

0

0

limx8

3

x 2x2 7x 8 = limx8

3

x 2x2 7x 8


213/289


Examples

Evaluate: limx8

3

x 2x2 7x 8

0

0

limx8

3

x 2x2 7x 8 = limx8

3

x 2x2 7x 8

3

x2


214/289


Examples

Evaluate: limx8

3

x 2x2 7x 8

0

0

limx8

3

x 2x2 7x 8 = limx8

3

x 2x2 7x 8

3

x2

+2 3

x


215/289


Examples

Evaluate: limx8

3

x 2x2 7x 8

0

0

limx8

3

x 2x2 7x 8 = limx8

3

x 2x2 7x 8

3

x2

+2 3

x+4


216/289


Examples

Evaluate: limx8

3

x 2x2 7x 8

0

0

limx8

3

x 2x2 7x 8 = limx8

3

x 2x2 7x 8

3

x2

+2 3

x+43x2 +2 3

x+4


217/289


Examples

Evaluate: limx8

3

x 2x2 7x 8

0

0

limx8

3

x 2x2 7x 8 = limx8

3

x 2x2 7x 8

3

x2

+2 3

x+43x2 +2 3

x+4

= limx8

x 8


218/289



219/289

Examples

Evaluate: limx8

3

x 2x2 7x 8

0

0

limx8

3

x 2x2 7x 8 = limx8

3

x 2x2 7x 8

3

x2

+2 3

x+43x2 +2 3

x+4

= limx8

x 8(x 8)(x+1)( 3

x2 +2 3

x+4)


220/289


Examples

Evaluate: limx8

3

x 2x2 7x 8

0

0

limx8

3

x 2x2 7x 8 = limx8

3

x 2x2 7x 8

3

x2

+2 3

x+43x2 +2 3

x+4

= limx8

x 8(x 8)(x+1)( 3

x2 +2 3

x+4)

1


221/289

= lim

x81

(x+1)( 3x2 +2 3x+4)


Examples

Evaluate: limx8

3

x 2x2 7x 8

0

0

limx8

3

x 2x2 7x 8 = limx8

3

x 2x2 7x 8

3

x2

+2 3

x+43x2 +2 3

x+4

= limx8

x 8(x 8)(x+1)( 3

x2 +2 3

x+4)

1


222/289

= lim

x81

(x+1)( 3x2 +2 3x+4)=

1

9(4+4+4)


Examples

Evaluate: limx8

3

x 2x2 7x 8

0

0

limx8

3

x 2x2 7x 8 = limx8

3

x 2x2 7x 8

3

x2

+2 3

x+43x2 +2 3

x+4

= limx8

x 8(x 8)(x+1)( 3

x2 +2 3

x+4)

1


223/289

= lim

x81

(x+1)( 3x2 +2 3x+4)=

1

9(4+4+4)

= 1

108


For today


2 Evaluating Limits


224/289

3 One-sided Limits


Illustration 4

Consider: f(x) =

3 5x2, x < 1

4x

3, x

1


225/289


Illustration 4

Consider: f(x) =

3 5x2, x < 1

4x

3, x

1

Asx 1, the value of f(x)dependson whetherx < 1or x > 1.


226/289


Illustration 4

Consider: f(x) =

3 5x2, x < 1

4x

3, x

1

Asx 1, the value of f(x)dependson whetherx < 1or x > 1. 4 3 2 1 1 2 3

1

2

3

4

0


227/289

3

2

1


Illustration 4

Consider: f(x) =

3 5x2, x < 1

4x

3, x

1


1

1

2

3

4

0


228/289

3

2

1

Asx approaches1 through values less than1,


Illustration 4

Consider: f(x) =

3 5x2, x < 1

4x

3, x

1


1

1

2

3

4

0


229/289

3

2

1

Asx approaches1 through values less than1, f(x)approaches 2.


Illustration 4

Consider: f(x) =

3 5x2, x < 1

4x

3, x

1


1

1

2

3

4

0


230/289

3

2

1

Asx approaches1 through values less than1, f(x)approaches 2.Asx approaches1 through values greater than1,


Illustration 4

Consider: f(x) =

3 5x2, x < 1

4x

3, x

1


1

1

2

3

4

0


231/289

3

2

1

Asx approaches1 through values less than1, f(x)approaches 2.Asx approaches1 through values greater than1, f(x)approaches1.


Illustration 5

Consider: g(x) =

x

2 1 1 2 3

1

2

0


232/289

1


Illustration 5

Consider: g(x) =

x

2 1 1 2 3

1

2

0

( )


233/289

1


Illustration 5

Consider: g(x) =

x

2 1 1 2 3

1

2

0

( )


234/289

1


Illustration 5

Consider: g(x) =

x

2 1 1 2 3

1

2

0

( )


235/289

1


Illustration 5

Consider: g(x) =

x

2 1 1 2 3

1

2

0

( )


236/289

1


Illustration 5

Consider: g(x) =

x

2 1 1 2 3

1

2

0

( )


237/289

1

Since there is no open interval Icontaining0 such thatg(x)is defined onI,


Illustration 5

Consider: g(x) =

x

2 1 1 2 3

1

2

0

( )


238/289

1

Since there is no open interval Icontaining0 such thatg(x)is defined onI, we

cannot letxapproach0 from both sides.


Illustration 5

Consider: g(x) =

x

2 1 1 2 3

1

2

0

( )


239/289

1

Since there is no open interval Icontaining0 such thatg(x)is defined onI, we

cannot letxapproach0 from both sides.

But we can say something about the values of g(x)as xgets closer and closer to0from the right of 0.


One-sided Limits

Intuitive Definition

The

limit of f(x)asx approachesa from the left is L


240/289


One-sided Limits


The


if the values of f(x)get closer and closer to Las the values of xget closer and

closer toa, but are less thana.


241/289


One-sided Limits


The


if the values of f(x)get closer and closer to Las the values of xget closer and

closer toa, but are less thana.


242/289

Notation:

limxa

f(x) = L


One-sided Limits


The

limit of f(x)asxapproachesafrom the right is L


243/289


One-sided Limits


The

limit of f(x)asxapproachesafrom the right is L

if t

Documents

M53 Lec1.1 Limits-OneSided1