Download 1.3 Some limit theorems

1.3 Some Limit Theorem
Theorem 2:
Let
px   c0  c1 x    cn x n be a polynomial, where c0 , c1 ,, cn
are real numbers and n is a fixed nonnegative integer. Then, for every real
x0 ,
number
lim px   px0   c0  c1 x0  c2 x02    cn x0n .
xx0
Theorem 3:
f x  and lim g x 
Let c be a real number, and suppose that both xlim
x
x x
0
exist. Then
cf x, lim  f x  g x, lim f xg x all exist.
(i) xlim
x
xx
x x
0
0
0
cf x   c lim f x  .
(ii) xlim
x
x x
0
0
 f x  g x  xlim
f x   lim g x 
(iii) xlim
x
x
x x
0
0
0
f x g x    lim f x   lim g x  .
(iv) xlim
 x x
  x x

x

0
0
g x   0 , then lim
(v) If xlim
x x0
x
0

0

f x 
exists and
g x 
f x 
f  x  xlim
 x0
lim

x  x0 g  x 
lim g  x  .
x  x0
Example 9:
By theorem 1,


lim x 3  2 x  6  33  2  3  6  27 .
x3
1
0
Example 10:
 x 2  1  by theorem2, (ii)
 x 2  1  by example 3

  3  2  6 .
lim 3

3 lim 
x1
x1
x

1
x

1




Example 11:
 x2 1
 by theorem2, (iii)  x 2  1 
3
  lim 4 x 3  3
lim 
 4 x  3 

lim 
x 1
x 1
 x 1

 x  1  x1
 2  4 13  3  9


Example 12:


lim x  3
2
x1
2
by theorem2, (iv)

since


x1



lim x 2  3  lim x 2  3  2  2  4

x1
lim x 2  3   1  3  2 .
x1
2
Example 13:




x3  x 2  3
 x3  x 2  3  by theorem2, (v) lim
45
x4

lim  2


 5,
2
x4 x  3x  5
lim
x

3
x

5
9


x4
since


lim x 3  x 2  3  43  42  3  45
x4
and


lim x 2  3x  5  42  3  4  5  9  0 .
x4
Example 14:


lim x  3
x1
2
10
by theorem2, (iv)







lim x 2  3  lim x 2  3  lim x 2  3
x1
x1
 2  2   2  1024
2
x1