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Introduction to Limits Section 1.2

Introduction to Limits Section 1.2. What is a limit?. A Geometric Example. Look at a polygon inscribed in a circle. As the number of sides of the polygon increases, the polygon is getting closer to becoming a circle. If we refer to the polygon as an n-gon ,

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Introduction to Limits Section 1.2

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  1. Introduction to LimitsSection 1.2

  2. What is a limit?

  3. A Geometric Example • Look at a polygon inscribed in a circle As the number of sides of the polygon increases, the polygon is getting closer to becoming a circle.

  4. If we refer to the polygon as an n-gon, where n is the number of sides we can make some mathematical statements: • As n gets larger, the n-gon gets closer to being a circle • As n approaches infinity, the n-gon approaches the circle • The limit of the n-gon, as n goes to infinity is thecircle

  5. The symbolic statement is: The n-gon never really gets to be the circle, but it gets close - really, really close, and for all practical purposes, it may as well be the circle. That is what limits are all about!

  6. FYI Archimedes used this method WAY before calculus to find the area of a circle.

  7. An Informal Description If f(x) becomes arbitrarily close to a single number L as x approaches c from either side, the limit for f(x) as x approaches c, is L. This limit is written as

  8. Numerical Examples

  9. Numerical Example 1 Let’s look at a sequence whose nth term is given by: What will the sequence look like? ½ , 2/3, ¾, 5/6, ….99/100, 99999/100000…

  10. What is happening to the terms of the sequence? ½ , 2/3, ¾, 5/6, ….99/100, 99999/100000… Will they ever get to 1?

  11. Numerical Example 2 Let’s look at the sequence whose nth term is given by 1, ½, 1/3, ¼, …..1/10000, 1/10000000000000…… As n is getting bigger, what are these terms approaching?

  12. Graphical Examples

  13. Graphical Example 1 As x gets really, really big, what is happening to the height, f(x)?

  14. As x gets really, really small, what is happening to the height, f(x)? Does the height, or f(x) ever get to 0?

  15. Graphical Example 2 As x gets really, really close to 2, what is happening to the height, f(x)?

  16. Graphical Example 3 Find

  17. Graphical Example 3 Use your graphing calculator to graph the following: Find As x gets closer and closer to 2, what is the value of f(x) getting closer to?

  18. Does the function exist when x = 2?

  19. ZOOM Decimal

  20. Limits that Fail to Exist

  21. Nonexistence Example 1: Behavior that Differs from the Right and Left What happens as x approaches zero? The limit as x approaches zero does not exist.

  22. Nonexistence Example 2 Discuss the existence of the limit

  23. Nonexistence Example 3: Unbounded Behavior Discuss the existence of the limit

  24. Nonexistence Example 4: Oscillating Behavior Discuss the existence of the limit

  25. Common Types of Behavior Associated with Nonexistence of a Limit

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