Rational Limit Forms
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Transcript of Rational Limit Forms
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Mr. Simonds MTH 251
R a t i o n a l L i m i t F o r m s | 1
Table 1: Ration al Limi t FormsForm Example Course of Action
real number
non-zero real number
3
2 16lim
5 4 x
x
x
+
(the form is22
11)
The value of the limit is the number to which the limit
form simplifies; for example,3
2 16lim 2
5 4 x
x
x
+ = .
You can immediately begin applying limit laws if you areproving the limit value
non-zero real numberzero
7
7lim
7 x
x
x
+
(the form is14
0
)
The limit value doesnt exist. The expression whoselimit is being found is either increasing without boundor decreasing without bound (or possibly both if youhave a two-sided limit). You may be able tocommunicate the non-existence of the limit using or . For example, if the value of x is a little less
than7
, the value of
7
7
x
x
+
is positive. Hence you
could write7
7lim
7 x
x
x
+ = .
zerozero
2
22
4lim3 2 x
x
x x
+ +
This is an indeterminate form limit . You do not knowthe value of the limit (or even if it exists) nor can youbegin to apply limit laws. You need to manipulate theexpression whose limit is being found until theresultant limit no longer has indeterminate form. Forexample:
( )( )( )( )( )( )
2
22 2
2
2 24lim lim
3 2 1 2
2lim
1
x x
x
x x x
x x x x
x
x
+ =+ + + +
=
+
(The limit form is now4
1
.)
real numberinfinity
( )01 4
limln
x
x
e
x +
(the form is3
)
The limit value is zero and this is justified by LimitLaw R3.
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