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Limits cont.

Limits cont. Evaluating them- Numerically, Anallyitically and Graphically. 1.3 Properties of Limits :. Properties :. Scalar multiple Sum or difference Product Quotient Power . Scalar multiple :.

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Limits cont.

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  1. Limits cont. Evaluating them- Numerically, Anallyitically and Graphically

  2. 1.3 Properties of Limits:

  3. Properties: • Scalar multiple • Sum or difference • Product • Quotient • Power

  4. Scalar multiple : • let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits: • let b = 2, c = 5, f(x) = x and g(x)=x2

  5. Sum or difference: • let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits: • let b = 2, c = 5, f(x) = x and g(x)=x2

  6. product: • let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits: • let b = 2, c = 5, f(x) = x and g(x)=x2

  7. quotient: • let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits: • let b = 2, c = 5, f(x) = x and g(x)=x2

  8. power: • let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits: • let b = 2, c = 5, f(x) = x and g(x)=x2

  9. Composition of Functions: • Functions can all be combined to form more complex functions: • We can also look at this another way: • Sometimes this makes it easier to calculate… we will deal with this more in the future.

  10. Trigonometric Limits: • Let c be a real number in the domain of the given trigonometric function: • This works this way for all of the trig functions.

  11. TrigonometrIc Limits:

  12. Finding Limits:Dividing Out Technique Consider Direct substitution yields an indeterminate form. Graph it. What does the function approach as x gets closer to 2? What about a table? What does it look like it is approaching? Can we do algebra? (see next slide for steps) What answer does that give us?

  13. Use algebra: Find: Use : So, = = 12

  14. Solving limits: • direct substitution. • Simplify using algebra and then try direct substitution again. • Use a graph or a table to reinforce your conclusion or to evaluate the limit if you are allowed to use technology.

  15. Squeeze theorem: • Not necessary for AP • to be added later

  16. Definition • Formal Definition: Epsilon-delta This formal definition is rather intimidating when you first look at it, but when broken down it makes sense.

  17. One-Sided limits: • Lets look at x3. • The limit from the left = ? • 1 • The limit from the right = ? • 4 • Are they equal? • NO • Does the limit exist? • NO!!!

  18. One-sided limits: • As we approach 3 from the left, we are approaching what value for the function? • What about from the right?

  19. Evaluating one-sided limits: Consider the following problem: When we do direct substitution, what do we get? We can do some algebra: First factor… Now calculate the limit:

  20. Another approach: What does it appear to be heading towards? Did we get the same value as the previous technique?

  21. Applications • Continuity • This is covered in the next section • Asymptotes • This is used in Curve Sketching (without a calculator!)

  22. Homework: • Pg 67 #6-40E

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