Indetermidiate forms
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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.
![0x Form
Limit of the form lim f(x) = 0, lim g(x)=
X0 X0
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 Hopitals rule.
14](https://image.slidesharecdn.com/indetermidiateforms-170507055604/85/Indetermidiate-forms-14-320.jpg)
![- Form
Limit of the form lim f(x) = , lim g(x)=
X0 X0
are called indeterminate form of the type -.
If we write form
lim [f(x)-g(x)] = lim [1/g(x)-1/f(x)] ,
X0 X0 1/[f(x)g(x)]
then the limit becomes of the form (0/0) & can be
evaluated by using the hopitals rule.
17](https://image.slidesharecdn.com/indetermidiateforms-170507055604/85/Indetermidiate-forms-17-320.jpg)
Indetermidiate forms
- 1. ACTIVE LEARNING PROCESS Indeterminate Forms Mechanical 1 Batch - C 1
- 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 2
- 3. History Of Indeterminate Form The term was originally introduced by Cauchys student Moigno in the middle of the 19th century 3
- 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. 4
- 5. Types of Indeterminate Forms There are seven types of indeterminate forms are as follow : 1. 0/0 2. / 3. 0* 4. - 5. 0 6. 1 7. 5
- 6. L Hopital 6 Actually, LHopitals Rule was developed by his teacher Johann Bernoulli. De lHopital paid Bernoulli for private lessons, and then published the first Calculus book based on those lessons. And has rights to use Bernoullis discoveries. Guillaume De l'H担pital 1661 - 1704
- 7. Johann Bernoulli 7 Johann Bernoulli 1667 - 1748 Johann Bernoulli was a Swiss Mathematician Johann was sent to LHopital 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 LHopitals rule on indeterminate forms.
- 8. LHopitals Rule L Hopitals 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) x0 x0 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. 8
- 9. LHospitals 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 Hopitals 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. 9
- 14. 0x Form Limit of the form lim f(x) = 0, lim g(x)= X0 X0 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 Hopitals rule. 14
- 17. - Form Limit of the form lim f(x) = , lim g(x)= X0 X0 are called indeterminate form of the type -. If we write form lim [f(x)-g(x)] = lim [1/g(x)-1/f(x)] , X0 X0 1/[f(x)g(x)] then the limit becomes of the form (0/0) & can be evaluated by using the hopitals rule. 17
- 20. Exponential Indeterminate Forms Exponential Indeterminate forms are 0, 1, . The is called an indeterminate form of the type 20 )( )(lim xg ax xf
- 26. Form Example 26
- 27. Form Example 27
- 28. 28
- 29. Thank You 29