Chapter 2
(Exam I: 2.12.5, 2.7)
0
0
tan
)
(
)
(
lim
0
x
x
x
f
x
f
m
x
x


=
→
Conjugate trick
•
You have (A+B) factor on top or bottom, involving a
•
Multiply top and bottom by (AB)
RAT F
 limits of rational functions can be found by substitution if the limit of the
denominator is not 0.
)
(
)
(
)
(
)
(
lim
C
Q
C
P
x
Q
x
P
c
x
=
→
Quotient Law
 limit of a quotient of two functions is the quotient of their limits, if the
limit of the denominator is not 0.
x
x
x
x
x
97
lim
5
3
lim
97
5
3
lim
3
+
=
+
→
2.3
(example in notes)
Consider a function f(x) and particular numbers x
0
and L such that f(x) is defined at least
in some open interval surrounding x=x
0.
We say that the limit of f(x) as x approaches x
0
is the number L and write:
L
x
f
x
x
=
→
)
(
lim
0
If for every number ε > 0 there exists a corresponding number δ > 0 such that for all x:
0 <
<

0
x
x
δ
<

⇒
L
x
f
)
(
ε
Indeterminate Limit Forms
0
0
,
∞
∞
,
∞
•
0
,
0
0
,
∞
1
Determinate Limit Forms
+
0
5
,
0
2
3
,
42
+
∞
,
8

∏
,
57
0
,
3
5
,
+
0
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 Fall '08
 SKILLINGS
 Calculus, Derivative, Limits, Rational Functions, lim, Limit of a function

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