1. What is a Limit-an simple approach .pdf
- 1. What is a Limit Ms. Chathuri Ranawaka (B.Sc.(Sp.) in Math. (Hons)(USJP−SL), M.Sc.(UoC−SL)) Sri Lanka Technology Campus
- 2. What is a limit ?
- 3. What is a limit ? • One of the most basic and fundamental ideas of calculus is limits.
- 4. What is a limit ? • One of the most basic and fundamental ideas of calculus is limits. • Limits allow us to look at what happens in a very, very small region around a point.
- 5. What is a limit ? • One of the most basic and fundamental ideas of calculus is limits. • Limits allow us to look at what happens in a very, very small region around a point. • Two of the major formal definitions of calculus depend on limits
- 6. The idea of Limit……. Consider finding the area of a circle :
- 7. The idea of Limit……. Consider finding the area of a circle : n = 3 n = 4 n = 5 ……………………………........ n = 12
- 8. The idea of Limits……. Consider finding the area of a circle : n = 3 n = 4 n = 5 ……………………………........ n = 12
- 9. The idea of Limits……. Let 𝒇 𝒙 = 𝒙𝟐 − 𝒙 + 𝟐. Fill the table. 𝒙 𝒇(𝒙) 1.0 1.5 1.8 1.9 1.95 1.99 1.995 1.999
- 10. The idea of Limits……. Let 𝒇 𝒙 = 𝒙𝟐 − 𝒙 + 𝟐. Fill the table. 𝒙 𝒇(𝒙) 1.0 2.000000 1.5 2.750000 1.8 3.440000 1.9 3.710000 1.95 3.852500 1.99 3.970100 1.995 3.985025 1.999 3.997001
- 11. The idea of Limits……. Let 𝒇 𝒙 = 𝒙𝟐 − 𝒙 + 𝟐. Fill the table. 𝒙 𝒇(𝒙) 1.0 2.000000 1.5 2.750000 1.8 3.440000 1.9 3.710000 1.95 3.852500 1.99 3.970100 1.995 3.985025 1.999 3.997001 𝒙 𝒇(𝒙) 3.0 2.5 2.2 2.1 2.05 2.01 2.005 2.001
- 12. The idea of Limits……. Let 𝒇 𝒙 = 𝒙𝟐 − 𝒙 + 𝟐. Fill the table. 𝒙 𝒇(𝒙) 1.0 2.000000 1.5 2.750000 1.8 3.440000 1.9 3.710000 1.95 3.852500 1.99 3.970100 1.995 3.985025 1.999 3.997001 𝒙 𝒇(𝒙) 3.0 8.000000 2.5 5.750000 2.2 4.640000 2.1 4.310000 2.05 4.152500 2.01 4.030100 2.005 4.015025 2.001 4.003001
- 13. The idea of Limits……. Let 𝒇 𝒙 = 𝒙𝟐 − 𝒙 + 𝟐. Fill the table. Guess what is 𝒇 𝒙 as 𝒙 approaches 2? 𝒙 𝒇(𝒙) 1.0 2.000000 1.5 2.750000 1.8 3.440000 1.9 3.710000 1.95 3.852500 1.99 3.970100 1.995 3.985025 1.999 3.997001 𝒙 𝒇(𝒙) 3.0 8.000000 2.5 5.750000 2.2 4.640000 2.1 4.310000 2.05 4.152500 2.01 4.030100 2.005 4.015025 2.001 4.003001
- 14. The idea of Limits……. Let 𝒇 𝒙 = 𝒙𝟐 − 𝒙 + 𝟐. Guess what is 𝒇 𝒙 as 𝒙 approaches 2? “the limit of the function 𝒇 𝒙 = 𝒙𝟐 − 𝒙 + 𝟐 as 𝒙 approaches 2 is equal to 4.”
- 15. The idea of Limits……. Let 𝒇 𝒙 = 𝒙𝟐 − 𝒙 + 𝟐. Guess what is 𝒇 𝒙 as 𝒙 approaches 2? “the limit of the function 𝒇 𝒙 = 𝒙𝟐 − 𝒙 + 𝟐 as 𝒙 approaches 2 is equal to 4.”
- 16. Definition of a Limit of a function f(x) f(x) (x,f(x)) (x,f(x)) L a x x x No matter how x approaches a, f(x) approaches L.
- 17. Definition of a Limit of a function If as x approaches a (without actually attaining the value a), f(x) approaches the number L, then we say that “L is the limit of f(x) as x approaches a”, f(x) f(x) (x,f(x)) (x,f(x)) L a x x x No matter how x approaches a, f(x) approaches L.
- 18. Definition of a Limit of a function If as x approaches a (without actually attaining the value a), f(x) approaches the number L, then we say that “L is the limit of f(x) as x approaches a”, and write f(x) f(x) (x,f(x)) (x,f(x)) L a x x x No matter how x approaches a, f(x) approaches L. lim ( ) x a f x L
- 19. Left & Right Hand Limits lim ( ) x a f x lim ( ) x a f x 2 -2 -5 5 0 lim 3 0 x x 0 lim 3 0 x x
- 20. Left & Right Hand Limits Ex: Consider the graph of 𝑓 𝑥 given below.
- 21. Left & Right Hand Limits Ex: Consider the graph of 𝑓 𝑥 given below. Find left and right hand limits of 𝑓 𝑥 as 𝑥 approaches 0.
- 22. Left & Right Hand Limits Ex: Consider the graph of 𝑓 𝑥 given below. Find left and right hand limits of 𝑓 𝑥 as 𝑥 approaches 0. lim 𝑥→0− 𝑓 𝑥 =? lim 𝑥→0+ 𝑓 𝑥 =?
- 23. Left & Right Hand Limits Ex: Consider the graph of 𝑓 𝑥 given below. Find left and right hand limits of 𝑓 𝑥 as 𝑥 approaches 0. lim 𝑥→0− 𝑓 𝑥 = 1 lim 𝑥→0+ 𝑓 𝑥 = 0
- 24. Limit of a function at a point A function 𝒇 𝒙 has a limit as 𝒙 approaches 𝒄
- 25. Limit of a function at a point A function 𝒇 𝒙 has a limit as 𝒙 approaches 𝒄 if and only if it has right and left hand limits
- 26. Limit of a function at a point A function 𝒇 𝒙 has a limit as 𝒙 approaches 𝒄 if and only if it has right and left hand limits and these one sided limits are equal:
- 27. Limit of a function at a point A function 𝒇 𝒙 has a limit as 𝒙 approaches 𝒄 if and only if it has right and left hand limits and these one sided limits are equal: Ex: 0 10 20 30 40 50 5 10 15 20 x Does the limit exist for this function as 𝑥 approaches 15 ?
- 28. Limit of a function at a point A function 𝒇 𝒙 has a limit as 𝒙 approaches 𝒄 if and only if it has right and left hand limits and these one sided limits are equal: Ex: 0 10 20 30 40 50 5 10 15 20 x Does the limit exist for this function as 𝑥 approaches 15 ? lim 𝑥→15− 𝑓 𝑥 = lim 𝑥→15+ 𝑓 𝑥 =
- 29. Limit of a function at a point A function 𝒇 𝒙 has a limit as 𝒙 approaches 𝒄 if and only if it has right and left hand limits and these one sided limits are equal: Ex: 0 10 20 30 40 50 5 10 15 20 x Does the limit exist for this function as 𝑥 approaches 15 ? lim 𝑥→15− 𝑓 𝑥 = 20 lim 𝑥→15+ 𝑓 𝑥 = 36
- 30. Limit of a function at a point A function 𝒇 𝒙 has a limit as 𝒙 approaches 𝒄 if and only if it has right and left hand limits and these one sided limits are equal: Ex: 0 10 20 30 40 50 5 10 15 20 x Does the limit exist for this function as 𝑥 approaches 15 ? lim 𝑥→15− 𝑓 𝑥 ≠ lim 𝑥→15+ 𝑓 𝑥
- 31. Limit of a function at a point A function 𝒇 𝒙 has a limit as 𝒙 approaches 𝒄 if and only if it has right and left hand limits and these one sided limits are equal: Ex: 0 10 20 30 40 50 5 10 15 20 x Does the limit exist for this function as 𝑥 approaches 15 ? lim 𝑥→15− 𝑓 𝑥 ≠ lim 𝑥→15+ 𝑓 𝑥 So, lim 𝑥→15 𝑓 𝑥 DNE
- 32. Limit of a function at a point Ex: Find the limits of following functions as 𝑥 approaches 1
- 33. Limit of a function at a point Ex: Find the limits of following functions as 𝑥 approaches 1 𝒇 𝒙 = (𝒙 − 𝟏)(𝒙 + 𝟏) (𝒙 − 𝟏) g 𝒙 = 𝒙 + 𝟏, 𝒙 ≠ 𝟏 𝟏, 𝒙 = 𝟏
- 34. Limit of a function at a point Ex: Find the limits of following functions as 𝑥 approaches 1 lim 𝒙→𝟏 𝒇(𝒙) = 𝟐 lim 𝒙→𝟏 𝒈(𝒙) = 𝟐 lim 𝒙→𝟏 𝒉(𝒙) = 𝟐
- 35. Limit of a function at a point Ex: Find the limits of 𝑓 𝑥 as 𝑥 approaches 1, 2, 3, and 4.
- 36. Limit of a function at a point Notice: Limit is a number. The limit can exist even when the function is not defined at a point or has a value different from the limit.