Indetermidiate forms

Seven types of indeterminate 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 Cauchy’s student Moigno in the middle of the 19 th 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 : 0/0 ∞ / ∞ 0* ∞ ∞ - ∞ 0⁰ 1 ∞ ∞ ⁰ 5

6. L Hopital 6 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 7 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. 8

9. L’ Hospital’s Rule Rules to evaluate 0/0 form : Check whether the limit is an indeterminate form. If it is not, then we cannot apply L’ Hopital’s rule. Differentiate f(x) and g(x) separately. If g’(x) ≠ 0, then the limit will exist. It may be finite, +∞ or -∞. If g’(x) = 0 then follow rule 4. Differentiate f’(x) and g’(x) separately. Continue the process till required value is reached. 9

10. 0/0 Form Example 10

11. 0/0 Form Example 11

12. ∞/∞ Form Example 12

13. ∞/∞ Form Example 13

14. x ∞ 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. 14

15. 15 15 x ∞ Form Example 15

16. x ∞ Form Example 16

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. 17

18. 18 ∞- ∞ Form Example 18

19. ∞-∞ Form Example 19

20. Exponential Indeterminate Forms Exponential Indeterminate forms are 0⁰, 1 ∞, ∞ ⁰. The is called an indeterminate form of the type 20

21. Steps To Evaluate 21

22. 0⁰ Form Example 22

23. 0⁰ Form Example 23

24. 1 ∞ Form Example 24

25. 1 ∞ Form Example 25

26. ∞⁰ Form Example 26

27. ∞⁰ Form Example 27

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29. Thank You 29