4.2 Introduction to Limits

Июль 22, 2022

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4.2 Introduction to Limits

Содержание

2. Lecture Outline
3. Introduction The concept of a limit is the fundamental building block on which most calculus concepts
4. Introduction There is no end to the natural numbers. Say the largest number you can imagine,
5. Infinitely large…
14. Try analysing the limits by considering which term “grows faster”.
17. Your turn!
18. Your turn!
19. Infinitely small…
20. Infinitely small…
21. Infinitely small…
23. Rules
27. Your turn!
28. Your turn!
29. So far, we have been dealing with limits of sequences, which, as you should recall, are
30. Limit Laws
33. Your turn!
34. Your turn!
35. Convergent and divergent sequences A sequence is convergent when it tends to a real number. If
36. Example 3:
40. Napier’s Number, e The mathematical constant e is a real, irrational and transcendental number approximately equal
41. A special convergent sequence
42. Example 5:
43. Solution:
45. Rules
46. Your turn!
47. Your turn!
48. Indeterminations
49. Example 6:
51. Example 7:
56. Your turn!
57. Your turn!
59. Example 10:
61. Learning outcomes After this lecture, you should be able to 4.2.1 Compute the limit of sequences;
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Lecture Outline

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Introduction
The concept of a limit is the fundamental building block on

which most
calculus concepts are based.
As you saw in the preview activity’s video, a group of atoms is so small that its size is almost zero. Except that it’s not zero. And the Local Galaxy Group is 10 million light years away from us. This distance so large that we find it hard to imagine. But we can still measure it.

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Introduction
There is no end to the natural numbers. Say the largest

number you can imagine, the colleague sitting next to you will say the same number plus one, and that will be a larger number. And we can go on and on forever…

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Infinitely large…

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Try analysing the limits by considering which term “grows faster”.

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Your turn!

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Your turn!

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Infinitely small…

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Infinitely small…

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Infinitely small…

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Rules

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Your turn!

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Your turn!

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So far, we have been dealing with limits of sequences, which,

as you should recall, are functions whose domain is the natural numbers.
It is also possible to compute limits of other functions, at any point in the domain of the function. It does not have to be always at infinity!

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Limit Laws

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Your turn!

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Your turn!

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Convergent and divergent sequences
A sequence is convergent when it tends to

a real number.
If a sequence tends to infinity or if it oscillates between two or more limits, then we say it is divergent.
A sequence cannot have two different limits.

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Example 3:

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Napier’s Number, e
The mathematical constant e is a real, irrational and

transcendental number approximately equal to:
e ≈ 2.71828 18284 59045 23536 02874 71352 66249 77572 47093 69995 95749 66967 62772 40766 30353 54759 45713 82178 53516 64274 27466 39193 20030 59921 81741 35966 29043 57290 03342 95260 59563 07381 32328 62794 34907 63233 82988 07531 95251 01901…

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A special convergent sequence

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Example 5:

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Solution:

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Rules

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Your turn!

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Your turn!

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Indeterminations

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Example 6:

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Example 7:

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Your turn!

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Your turn!

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Example 10:

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Learning outcomes
After this lecture, you should be able to
4.2.1 Compute

the limit of sequences;
4.2.2 Determine whether a sequence is convergent or divergent;
4.2.3 Compute limits of other functions numerically and algebraically.