1

Section 2.4One-Sided Limits and Limits at Infinity

ال ما عند والنهايات الجانب أحادية النهاياتنهاية

2

0 if 1

0 if 1)(

xx

x

xx

x

x

xxf

3

بشكل حدسي

4

5

6

7

x sin(1/x)0.1 -0.54402111090.01 -0.50636564110.001 0.82687954050.0001 -0.30561438890.00001 0.03574879800.000001 -0.34999350220.0000001 0.42054779320.00000001 0.9316390271

-0.1 0.54402111090.01- 0.5063656411-0.001 -0.82687954050.0001- 0.30561438890.00001- -0.03574879800.000001- 0.34999350220.0000001- -0.42054779320.00000001- 0.9316390271-

مَت�ُذ�بُذبة

8

9

x 1/x10 0.1

100 0.011000 0.001

10000 0.0001100000 0.00001

1000000 0.00000110- 0.1-

100- 0.01-1000- 0.001-

10000- 0.0001-100000- 0.00001-

1000000- 0.000001-

10

11

12

.c lim)...(c lim (b)

.c lim)...(c lim (a)

then ,0 if , as spolynomial of Limits

011

1

011

1

nn

x

nn

nn

x

nn

x

nn

nn

x

n

xcxcxcx

xcxcxcx

cx

)9247( lim : Example 35

xxxx

even isk and 0 if

odd isk and 0 if

even isk and 0 if

odd isk and 0 if

lim

0 if

0 if lim

c

c

c

c

cx

c

ccx

k

x

k

x

)9247( lim : Example 35

xxxx

57 lim xx

57 lim xx

13

14

15

A QUICK METHOD FOR FINDING LIMITS OF RATIONAL FUNCTIONS AS X→+∞ OR X -∞ →

.lim...

...lim

)(

)(lim)(lim

then, ...

...

)(

)()(

such that functions polynomial be q(x)andp(x)Let

011

1

011

1

011

1

011

1

nn

mm

xnn

nn

mm

mm

xxx

nn

nn

mm

mm

xb

xa

bxbxbxb

axaxaxa

xq

xpxf

bxbxbxb

axaxaxa

xq

xpxf

.nm if,limlimlim)(

)(lim)(lim (b)

n.m if ,

nm if ,0lim

)(

)(lim)(lim (a)

nm

xn

mnm

n

m

xnn

mm

xxx

n

mnn

mm

xxx

xb

ax

b

a

xb

xa

xq

xpxf

b

axb

xa

xq

xpxf

.31

123lim (c),

86

53lim (b),

52

4lim (a)

limits following theCompute : Example23

3

2

x

xx

x

x

x

xxxxx

16

.31

125 lim Find : Example

23

x

xxx

17

The graph appears to approach the horizontal line y = 0, as x →+∞and as x →−∞. In this case, we call y = 0 a horizontal asymptote.

18

19

-1.0 -0.5 0.5 1.0 1.5 2.0 2.5 3.0

-1

1

2

3

x

y

End of the section

20

SECTION 1.5 LIMITS INVOLVING INFINITY; ASYMPTOTES

وخطوط النهاية ما المَتضمنة النهاياتالَتقارب

When this occurs, we say that the line x = 0 is a vertical asymptote.

we say that the line x = 5 is a vertical asymptote.

we say that the line x = -2 and x=3 are vertical asymptotes.

سهلة) نهايةمعادة(

21

x sin(1/x)0.1 -0.54402111090.01 -0.50636564110.001 0.82687954050.0001 -0.30561438890.00001 0.03574879800.000001 -0.34999350220.0000001 0.42054779320.00000001 0.9316390271

-0.1 0.5440211109-0.01 0.5063656411-0.001 -0.8268795405-0.0001 0.3056143889-0.00001 -0.0357487980-0.000001 0.3499935022-0.0000001 -0.4205477932-0.00000001 -0.9316390271

x sin(1/x)0.1 -0.54402111090.01 -0.50636564110.001 0.82687954050.0001 -0.30561438890.00001 0.03574879800.000001 -0.34999350220.0000001 0.42054779320.00000001 0.9316390271

-0.1 0.54402111090.01- 0.5063656411-0.001 -0.82687954050.0001- 0.30561438890.00001- -0.03574879800.000001- 0.34999350220.0000001- -0.42054779320.00000001- 0.9316390271-

22

23

24

25

26

27

28

29