Limit of a Function and One-sided Limits

8/12/2019 Limit of a Function and One-sided Limits

1/289

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


Ill t ti 1


24/289

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 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


27/289

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)

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


31/289

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


32/289

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

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


38/289


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




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Illustration 3


43/289

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




47/289

Illustration 3


48/289

Consider: h(x) =

3x 1, x =10, x=1

1 1 2 31

1

2

3

4



Illustration 3


49/289

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


51/289

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


57/289

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


59/289

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


61/289

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


69/289

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).



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Some Remarks


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Remark

In finding limxa f(x):

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


76/289

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)



101/289

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



107/289

Evaluate: limx1

(2x2 +3x 4)

limx1

(2x2 +3x 4)


108/289


Evaluate: limx1

(2x2 +3x 4)

limx1

(2x2 +3x 4) = limx1

2x2


109/289



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


Evaluate: limx1

(2x2 +3x 4)

limx1

(2x2 +3x 4) = limx1

2x2 + limx1

3x limx1

4

2

li2

3

li

li 4


124/289

=2

limx1 x

+3

limx1 x limx1 4

=2

limx1

x

2+3

limx1

x

lim

x14

=2(

1)2 +3(

1)

4

= 5


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



144/289

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



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



159/289

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,





Note that limx1

3x2 4x+1

=0and lim

x1(x 1) = 0.


170/289


x

1 =

(3x 1)(x 1)x

1 =3x 1.


limx1

3x2 4x+1x 1





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


Examples

Evaluate: lim

x1

x2 +2x+1

x+1

0

0lim

x1x2 +2x+1

x + 1 = lim

x1(x+1)2


185/289

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


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


Examples

Evaluate: limx

2

x3 +8

x2

4


190/289


Examples

Evaluate: limx

2

x3 +8

x2

4

0

0


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



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


Examples

Evaluate: limx4

x2 162x

0

0

limx4x2

16

2x = limx4x2

16

2x 2+

x

2+x

= limx4

(x2 16)(2+x)4 x


206/289

= limx4

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


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


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)


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

8/12/2019

Limit of a Function and One-sided Limits

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