This document provides an overview of indeterminate forms in calculus and L'Hopital's rule for evaluating limits that result in indeterminate forms. It discusses the seven types of indeterminate forms: 0/0, ∞/∞, 0×∞, ∞-∞, 00, 1∞, and ∞0. For each type, it outlines the process for applying L'Hopital's rule to differentiate the numerator and denominator and evaluate the limit. The document also provides background on L'Hopital and Johann Bernoulli, who originally developed the rule, and includes examples of applying L'Hopital's rule to limits in various indeterminate forms.

1. ACTIVE LEARNING
PROCESS
Indeterminate Forms
Mechanical – 1
Batch - C
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2. Calculus (2110014)
Indeterminate Forms
Guided by :
Asst. Prof. Bhavesh Suthar Created by:
Preet Shah – 160410119117
Hitesh Rawal – 160410119111
Dishant Vaidhya – 160410119135
Chirag Kataria – 160410119030
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3. History Of Indeterminate Form
• The term was originally
introduced by Cauchy’s
student Moigno in the
middle of the 19th century
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4. Indeterminate Forms
• In calculus and other branches of mathematical
analysis, limit involving an algebraic
combination of function in an independent
variable may often be evaluated by replacing
these function by their limits.
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5. Types of Indeterminate Forms
 There are seven types of indeterminate forms are as
follow :
1. 0/0
2. ∞ / ∞
3. 0* ∞
4. ∞ - ∞
5. 0⁰
6. 1∞
7. ∞⁰
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6. L Hopital
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Actually, L’Hopital’s Rule
was developed by his
teacher Johann Bernoulli.
De l’Hopital paid
Bernoulli for private
lessons, and then published
the first Calculus book
based on those lessons.
And has rights to use
Bernoulli’s discoveries.
Guillaume De l'Hôpital
1661 - 1704

7. Johann Bernoulli
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Johann Bernoulli
1667 - 1748
Johann Bernoulli was a Swiss Mathematician
Johann was sent to L’Hopital in Paris
to teach a method or rule for solving
problems involving limits that would
apparently be expressed by the ratio of
zero to zero, now called L’Hopital’s rule on
indeterminate forms.

8. L’Hopital’s Rule
• L’ Hopital’s Rule is a general method for evaluating the
indeterminate forms 0/0 and ∞/∞. This rule states that
lim f(x)/g(x) = lim f’(x)/g’(x)
x→0 x→0
where f’ & g’ are the derivatives of f & g.
• Note : This rule does not apply to expression ∞/0 and 1/0, & so on.
• These derivatives will allow one to perform algebraic simplification and
eventually evaluate the limit.
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9. L’Hospital’s Rule
• 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’(x) ≠ 0, then the limit will exist. It may be
finite, +∞ or -∞. If g’(x) = 0 then follow rule 4.
4. Differentiate f’(x) and g’(x) separately.
5. Continue the process till required value is
reached.
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10. 0/0 Form Example
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11. 0/0 Form Example
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12. ∞/∞ Form Example
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13. ∞/∞ Form Example
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14. 0x∞ Form
• Limit of the form lim f(x) = 0, lim g(x)=∞
X→0 X→0
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.
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15. 1515
0x∞ Form Example
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16. 0x∞ Form Example
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17. ∞-∞ Form
• Limit of the form lim f(x) = ∞, lim g(x)=∞
X→0 X→0
are called indeterminate form of the type ∞-∞.
• If we write form
• lim [f(x)-g(x)] = lim [1/g(x)-1/f(x)] ,
X→0 X→0 1/[f(x)∙g(x)]
then the limit becomes of the form (0/0) & can be
evaluated by using the hopital’s rule.
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18. 18
∞-∞ Form Example
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19. ∞-∞ Form Example
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20. Exponential Indeterminate Forms
• Exponential Indeterminate forms are 0⁰, 1∞, ∞⁰.
• The is called an indeterminate form of the
type
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)(
)(lim
xg
ax
xf


21. Steps To Evaluate
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22. 0⁰ Form Example
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23. 0⁰ Form Example
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24. 1∞ Form Example
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25. 1∞ Form Example
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26. ∞⁰ Form Example
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27. ∞⁰ Form Example
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29. Thank You
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