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GISAT 151 (Week 4a)

GISAT 151 (Week 4a). § 3.6 Limits: Numerical and Graphical Approaches. “ The derivative is defined using a limit , and it is now time to say more precisely what that means. It is possible to speak of limits by themselves, rather than in the context of the derivative…..”.

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GISAT 151 (Week 4a)

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  1. GISAT 151 (Week 4a) §3.6 Limits:Numerical and Graphical Approaches “The derivative is defined using a limit, and it is now time to say more precisely what that means. It is possible to speak of limits by themselves, rather than in the context of the derivative…..” as xapproaches a from both sides (x- and x+). If not approaches same number L, then its limit does not exist (or no limit)

  2. Evaluating Limits Numerically Ex. Given the function f (x) = 2x2 – 3, what happens to f as x approaches 2? As x gets closer to2, f gets closer to 5.

  3. Limit of a Function If f (x) approaches the number L as x approaches a from both sides we say that the limit of f equals L and write: If f (x) fails to approach a single number as x approaches a from both sides we say that fhas no limit:

  4. One-Sided Limits of a Function Ex. Given Find Find

  5. Limits at Infinity Ex. Let x get arbitrarily large. f approaches

  6. Limits at Infinity Problem 17 (§ 3.6): Let x get arbitrarily large. xe-xapproaches 0

  7. Evaluating Limits Graphically • Draw the graph of f. • Move along the graph toward x = a from the rightand read the y coordinate (limit). • Move along the graph toward x = a from the left and read the y coordinate (limit). • If the left and right limits both exist and are the same value L, then

  8. Evaluating Limits Graphically • To evaluate • Move along the graph to the far right and read the y coordinate (limit) that the function approaches (if any). • To evaluate • Move along the graph to the far left and read the y coordinate (limit) that the function approaches (if any).

  9. Computing Limits Graphically Ex. 6 Note: f (-2) = 1 is not involved • 2

  10. Computing Limits Graphically Ex. 2 5 -2 Notice the limit from one side of 5 is 2 and from the other side of 5 is –2.

  11. Computing Limits Graphically – Infinite Limit Ex. 6 • 2

  12. §3.6 Problem 32: Airline Stocks Prior to the September 11, 2001, attacks, United Airlines stock was trading at around $35/share, Immediately following the attacks, the share price dropped by $15. Let U(t) be this cost at time t and t = 11 represent September 11, 2001. What does the given information tell you about Answer:

  13. §3.6 Hw Problem 41: What is wrong with the following statement? “Since f(a) is not defined, doest not exist.” Hints:

  14. L’Hospital’s Rule: If f and g are two differentiable functions such that substituting x = a in the expression f(x)/g(x) gives either 0/0 or /, then Examples:

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