Post on 03-Jan-2016
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Section 2.4One-Sided Limits and Limits at Infinity
ال ما عند والنهايات الجانب أحادية النهاياتنهاية
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0 if 1
0 if 1)(
xx
x
xx
x
x
xxf
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بشكل حدسي
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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
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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-
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.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
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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
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123lim (c),
86
53lim (b),
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4lim (a)
limits following theCompute : Example23
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2
x
xx
x
x
x
xxxxx
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.31
125 lim Find : Example
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x
xxx
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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.
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-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
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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(
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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-
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