This preview shows page 1. Sign up to view the full content.
Definition
: If
f
(
x
) becomes closer and closer to a single number
L
as
x
gets closer and closer
to
c
from either side, then
lim
x
→
c
f
(
x
) =
L,
which is read as “the limit of
f
(
x
) as
x
approaches
c
is
L
.”
Examples:
(
a
) lim
x
→
1
3
x
2
+ 5 = 3
×
(

1)
2
+ 5 = 8
.
(
b
) lim
x
→
2
p
x
2
+ 1 =
p
2
2
+ 1 = 5
(
c
) lim
x
→
0
1
x
2
+ 1
=
1
0
2
+ 1
= 1
.
(d) Suppose that
f
(
x
) =
±

x

x
6
= 0
1
x
= 0
.
Then
lim
x
→
0
f
(
x
) = 0
.
Note
:
lim
x
→
c
f
(
x
)
relies on the values of
f
(
x
) at
x
near
c
, but may not have any connection to the value of
f
(
x
)
at
x
=
c
.
Replacement Theorem
: If
f
(
x
) and
g
(
x
) are equal at all points except for at
x
=
c
. Then,
lim
x
→
c
f
(
x
) = lim
x
→
c
g
(
x
)
Example:
√
x
+ 1

1
This is the end of the preview. Sign up
to
access the rest of the document.
Unformatted text preview: x = √ x + 11 x × √ x + 1 + 1 √ x + 1 + 1 = x + 11 x ( √ x + 1 + 1) = 1 √ x + 1 + 1 as long as x 6 = 0 (why?). Therefore, lim x → √ x + 11 x = lim x → 1 √ x + 1 + 1 = lim x → 1 lim x → ( √ x + 1 + 1) = 1 2 Extra Problems: Find the following limits ( a ) lim x → x 4 + 3 x 35 x 2 x , ( b ) lim x → 1 x 21 x1 ( c ) lim x → 1 x1 x 21 ( d ) lim x → 1 x1 5 x5...
View
Full
Document
This note was uploaded on 11/21/2010 for the course MAT 16A 16A taught by Professor Xiao during the Fall '10 term at UC Davis.
 Fall '10
 Xiao

Click to edit the document details