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GyanmanriInstitute of Technology
(Bhavnagar)
Name of the student Roll no Enrollment number
Ronak Sutariya 48 151290107052Virav rakholiya. 55 151290107062
Shraddha sheladiya. 54 151290107008Nidhi mandaviya 44 151290107014
Sub: Calculus
Created by :Ronak Sutariya
Inderminate forms and improper integrals
objectives Recognize limits that produce indeterminate forms.
Apply L’Hôpital’s Rule to evaluate a limit.
Indeterminate FormsThe forms 0/0 and are called indeterminate because they do not guarantee that a limit exists, nor do they indicate what the limit is, if one does exist.
When you encountered one of these indeterminate forms earlier in the text, you attempted to rewrite the expression by using various algebraic techniques.
Indeterminate Forms
Occasionally, you can extend these algebraic techniques to find limits of transcendental functions. For instance, the limit
produces the indeterminate form 0/0. Factoring and then dividing produces
Indeterminate FormsNot all indeterminate forms, however, can be evaluated by algebraic manipulation. This is often true when both algebraic and transcendental functions are involved. For instance, the limit
produces the indeterminate form 0/0.Rewriting the expression to obtain
merely produces another indeterminate form,
Indeterminate Forms
You could use technology to estimate the limit, as shown in the table and in Figure given belove From the table and the graph,the limit appears to be 2.
L’Hospital’s RuleTo find the limit illustrated in given figure, you can use a theorem called L’Hospital’s Rule. This theorem states that under certain conditions, the limit of the quotient f(x)/g(x) is determined by the limit of the quotient of the derivatives
Example 1 – Indeterminate Form 0/0
Evaluate
Solution:Because direct substitution results in the indeterminate form 0/0.
Example 1 – Solution
You can apply L’Hospital’s Rule, as shown below.
L’Hospital’s RuleThe forms have been identified as indeterminate. There are similar forms that you should recognize as “determinate.”
As a final comment, remember that L’Hôpital’s Rule can be applied only to quotients leading to the indeterminate forms 0/0 and
Example
IMPROPER INTEGRALS
Improper Integral
TYPE-I:Infinite Limits of Integration
1 2 1 dxx
Example
1
1 2 1 dxx
TYPE-II:Discontinuous IntegrandIntegrands with Vertical
Asymptotes
IMPROPER INTEGRALS
DEFINITION OF AN IMPROPER INTEGRAL OF TYPE 1
b
aba dxxf dxxf )(lim)(
Example
1 2 1 dxx
IMPROPER INTEGRALS
DEFINITION OF AN IMPROPER INTEGRAL OF TYPE 1
b
aa
b dxxf dxxf )(lim)(
Example
0 dxxex
IMPROPER INTEGRALS
DEFINITION OF AN IMPROPER INTEGRAL OF TYPE 1
b
aa
b dxxf dxxf )(lim)(
b
aba dxxf dxxf )(lim)(
The improper integrals
are called convergent if the corresponding limit exists and divergent if the limit does not exist.
a dxxf )(
a dxxf )(
IMPROPER INTEGRALS
DEFINITION OF AN IMPROPER INTEGRAL OF TYPE 1
t
ata dxxf dxxf )(lim)(
Example
1 1 dx
x
IMPROPER INTEGRALS
DEFINITION OF AN IMPROPER INTEGRAL OF TYPE 1
dxxf )(
If both improper integrals
are convergent
a dxxf )(
a dxxf )(
convergent
dxxf )(
a
a dxxf dxxf )()(
Example
dx
x 11
2
DEFINITION OF AN IMPROPER INTEGRAL OF TYPE 2
Example
5
2
21 dx
x
Thank you
Created by :Ronak Sutariya
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