Calculus and Analytical Geometry

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Calculus and Analytical Geometry
MTH 104
Lecture 11
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LHÔPITALS RULE INDETERMINATE FORMS
INDETERMINATE FORMS OF TYPE
.
Limit of the form
in which
and
is called an indeterminate form of type
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LHopital Rule for form 0/0
Suppose that
and
are differentiable functions on an open
, and that
, except possible at
interval containing

, then
If
exists, or if this limit is
or
Moreover this statement is also true in the case
of limits as
or as
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EXAMPLE Find the limit
Using LHÔPITALS rule, and check the result by
factoring.
solution
form
Using LHÔPITALS rule
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By computation
Example
In each part confirm that the limit is an
indeterminate form of type 0/0 and evaluate it
using LHOPITALs rule
(a)
(b)
(c)
(d)
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(e)
(f)
Solution
(a)
Applying LHÔPITALS rule
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(b)
Applying LHÔPITALS rule
(c)
Applying LHÔPITALS rule
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(d)
Applying LHÔPITALS rule
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INDETERMINATE FORMS OF TYPE
The Limit of a ratio, in which
the numerator has limit and the denominator
has the limit is called an indetreminate form
of type

LHopital Rule for form
Suppose f and g are differentiable functions on
an open interval conatining xa, except possibly
at, xa and that
and
, then
or
exists, or if this limit is
If
Moreover this statement is also true in the case
of limits as
or as
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Example
In each part confirm that the limit is
indeterminate form of type

and apply LHÔPITALS
(b)
(a)
solution
(a)
Applying LHÔPITALS rule
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(b)
Applying LHÔPITALS rule
Any additional application of LHÔPITALS rule
will yield powers of
in the numerator and expressions involving
and
in the denominator.
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Rewriting last expression
Thus,
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INDETERMINATE FORMS OF TYPE
The limit of an expression that has one of the
forms


is called and indeterminate form if the limits
and individually exert conflicting
influences on the limit of the entire expression.
For example
Indeterminate form
On the other hand
Not an indeterminate form
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Indeterminate form of type
can sometimes be evaluated by rewriting the
product as a ratio, and then applying LHÔPITALS
rule for indeterminate form of type or
.

Example
Evaluate
(b)
(a)
Solution
(a)
Rewriting
form
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Applying LHÔPITALS rule
(b)
Rewriting as
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Applying LHÔPITALS rule


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Indeterminate forms of type
A limit problem that leads to one of the
expressions
.





is called an indeterminate form type
The limit problems that lead to one of the
expressions
are not indeterminate, since two terms work
together.
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Example
Evaluate
Solution
Rewriting
Applying LHÔPITALS rule
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Again Applying LHÔPITALS rule
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INDETERMINATE FORM OF TYPE
Limits of the form

can give rise to indeterminate forms of the types
and
For example
It is indeterminate because the expressions
and gives
--Two conflicting
influences. Such inderminate form can be
evaluated by first introducing a dependent
variable
The limit of lny will be an indeterminate form
of type
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Example

Solution Let
Applying LHÔPITALS rule
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