Lesson 17: Indeterminate Forms and L'Hopital's Rule (Section 041 slides)
Indeterminate Forms and L' Hospital Rule
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Transcript of Indeterminate Forms and L' Hospital Rule
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Indeterminate Forms
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By
Ketul Vaidya(15BEMEG061)Prashant Ranade(15BEMEG062)
Aakash Singh(15BEMEG063)Aditya(15BEMEG064)
Swapnil Bodke(15BEMEG065)
F.Y. B.E. MechanicalM – 1 Batch – C
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The term was originally introduced by Cauchy’s student Moigno in the middle of the 19th century.
History
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What are indeterminate forms?
◦ In calculus and other branches of mathematical analysis, limits involving an algebraic combination of functions in an independent variable may often be evaluated by replacing these functions by their limits.
◦ If the expression obtained after this substitution does not give enough information to determine the original limit, it is said to take on an indeterminate form.
Indeterminate Forms
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There are seven types of indeterminate forms :
1. 0/02. ∞/∞3. 0 × ∞4. ∞ − ∞5. 00
6. 1∞
7. ∞0
Types of Indeterminate forms
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Limit of the formare called indeterminate form of the type 0/0.
0/0 Form
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L'Hopital's rule is a general method for evaluating the indeterminate forms 0/0 and ∞/∞. This rule states that (under appropriate conditions)
where f' and g' are the derivatives of f and g. Note that this rule does not apply to expressions ∞/0, 1/0,
and so on. These derivatives will allow one to perform algebraic
simplification and eventually evaluate the limit.
L’ Hopital’s Rule
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Rules to evaluate 0/0 form :1. Check whether the limit is an indeterminate form. If it is
not, then we cannot apply L’ Hopital’s rule.2. Differentiate f(x) and g(x) separately.3. If g’(a) ≠ 0, then the limit will exist. It may be finite, +∞
or -∞. If g’(a)=0 then follow rule 4.4. Differentiate f’(x) & g’(x) separately.5. Continue the process till required value is reached.
L’ Hopital’s Rule
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If , then it is indeterminate form of type 0/0.
∞/∞ Form
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Limit of the form are called indeterminate form of the type 0x∞.
If we write f(x) g(x) = f(x)/[1/g(x)], then the limit becomes of the form (0/0).
This can be evaluated by using L’ Hopital’s rule.
0x∞ Form
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Example
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Limit of the form are called indeterminate form of the type ∞-∞.
If we write , then the limit becomes of the form (0/0) and can be evaluated by using the L’ Hopital’s rule.
∞-∞ Form
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Example
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Limit of the form are called indeterminate form of the type .
If we write , then the limit becomes of the form (0/0) and can be evaluated by using the L’ Hopital’s rule.
Form
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Example
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Limit of the form are called indeterminate form of the type .
If we write , then the limit becomes of the form (0/0) and can be evaluated by using the L’ Hopital’s rule.
Form
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Example
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Limit of the form are called indeterminate form of the type .
If we write , then the limit becomes of the form (0/0) and can be evaluated by using the L’ Hopital’s rule.
Form
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Example
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