x
lim
n
2n
sin
x 2n
(x
0)
x2 sin 1
lim
x
x 2x2 1
d.
Example
1
lim
x0
2x
1
1
2x 1
lim x x x
x
x 1
lim x 3sin x x 3x 2 cos x
e.
Example
lim( x2 x x)
x
lim( 1 1 ) x1 ln x x 1
3sin kx 2
1.If lim x0
2x
, 3
then
k=________.
A. 1 B. 3 2
C. 2 3
1
2.Iflim (1 kx) x
1,
then
x0
e
D. 4 9
k=________.
A. 1 B. 1
C. 2
D. 2
四、 Fill in the blanks:
1. lim x
x)
a 2
f (0 ) lim ln (b x2) ln b
a
x0
1 ln
b
2
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Example If has an infinite discontinuity at and has a removable discontinuity at Find a and b .
Example
lim 3 x 1 x1 x 1
lim x a x a
xa
x2 a2
lim
x0
1 tan x x3
1 sin x
c.
sin x lim 1 x0 x
0 0 lim sin x 0
x x
tan 6x
Example lim
x0 sin 2x
1
lim x sin 0
x0
x0
x
=________.
8. If lim (3x)x 3 1 x1 a(x 1)
,then a=________.
Derivatives 一、The chain rule Example Differentiate the function
1. f (x) x ln(x x2 1)
2. f (x) x arctan x
then there exists a number c between 1 and 3 such that
f (c) 0.
True or false
10.If f (x) 1 for all x and limx0 f (x) exists, then limx0 f (x) 1.
三、Choose the best answer for each of the Following
sin x
(x )ex
11 2.limx0 (x sin 2x x sin 2x)
3.limx0 x 1 2x
1
4.lim x0 e x
Example If
is continuous at x = 0 , then a = 2 , b = e .
f
(0 )
lim
x0
a (1 cos x2
7. If limx0 f (x) and limx0 g(x) , then limx0[ f (x) g(x)] 0.
Example limx[x x]
True or false
8.If limxa f 2 (x) exists
then limxa f (x) exists。
9. If f (1) 0 and f (3) 0,
lim(x ln x)
x
lim x ln x x x
Example lim ( x x x x)
x
f. 0
Example
2x
lim
x
x sin
x2
1
g. 1
lim(1 1 )x e
x
x
Example
lim( x 3)x x x 6
lim tann ( 1)
n
4n
lim(1
x 0
x 0
has a jump discontinuity at x = 0 .
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Example Find
Solution
lim
x0
2 1
1
ex
4
ex
sin x
x
lim
x0
2
e
4 x
e
e
4 x
1
3 x
sin x
x
1
lim
x0
2 1
1
ex
4
ex
sin x
lim f (x) 3 .
x
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六、 Example Show that there is a root of the equation
ex2 x
between 0 and 1.
Solution Let g(x) ex2 x, We have g(0) 1 0 g(1) e1 1 0
2.
lim
x2
6x
7
lim(x2
x1
6x
7)
x1 x2 5x 6 lim(x2 5x 6)
x1
3.
lim
x1
x2
x
3 2x
4
lim( x
x1
lim(x2
3) 2x
4)
x1
True or false
4.If limx5 f (x) 0 and limx5 g(x) 0,
then
That is ec2 c.
六、 Example
Prove that the equation sin x x
has exactly one real root。
Fill in the blanks:
1.If f (x0 ) 1,then
lim f (x0 3h) f (x0 )
h0
h
Review
Limits
一、f (x) is continuous at x0
a.
lim
xx0
f
(x)
f
(x0 )
Example
lim
x2
x2 2x 3 x4 3x
f
(2)
22 2 2 3 24 3 2
1 2
二
0 , , , 0 ,1 , 00, 0, 00. 0
b. 0 0
ex b
So,
lim
x1
x (x 1)
does not exist.
lim(ex b) 0
x1
b limex e
x1
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Example Find a , b , such that
Solution
Since
thus
lim ( 3
x
1 x3
1
a
b x
)
0
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Solution Because
and the value of
If
we know that the limit
does not exist,
it follows that the value of namely,
must be 0,
so
True or false
1. lim( 2x 8 ) lim 2x lim 8 x4 x 4 x 4 x4 x 4 x4 x 4
therefore 1 a 0 , a 1 ,
lim
1
x 3 (1 x3)2 x 3 1 x3 x2
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Example Find the points of discontinuity and identify its type, where
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x
(1) sin x 1 sin x
2sin x 1 x cos x 1 x
2
2
2sin
1
cos x 1 x
2( x 1 x)
2
无穷小
有界
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Example
Is there a number such that
exists? If so, find the value of the limit.
x)
x
2
sin
1 x
,
ax b,
x0 x0
Find the values of a and b that make f (x)
differentiable everywhere .
Solution
Because f is differentiable everywhere ,thus f is continuous and differentiable at x=0
lim f (x0 h) f (x0 h)
h0
h
=________.
Fill in the blanks:
5.Iff (x0 ) 1,then
lim f (x0 ah) f (x0 bh) =________.
h0
h
6.If f (0) 0 and f (0) 1, then
lim f (2x) f (3x) =________.