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2.1 Quiz

2.1 Quiz. Ch 2.2. Limits Involving Infinity. Warm Up (on noteguide this time) How do we find Horizontal Asymptotes of Simple Rational Functions of the form y = ?. If _________ then the line y=0 is the horizontal asymptote

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2.1 Quiz

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  1. 2.1 Quiz

  2. Ch 2.2 Limits Involving Infinity

  3. Warm Up (on noteguide this time) How do we find Horizontal Asymptotes of Simple Rational Functions of the form y = ? If _________ then the line y=0 is the horizontal asymptote If _________then the line y = is the horizontal asymptote If ________then it has no horizontal asymptote – it has a slanted asymptote, which you find using division m < n m= n m> n

  4. Warm Up CONT: Review from PRECALC /ALG 2 Graph each of the following. State any VAs, HAs, holes, x-ints, and y-ints • f(x)= b) g(x) = c) h(x) =

  5. PC Review: What other patterns are there in rational expressions? f(x) = vs g(x) = vs h(x) = vs f(x) =

  6. Why do we care about what happens to a function at Infinity? • We use infinity () to describe behavior of a function when the values in its domain or range outgrow all finite bounds. • Many people like to make predictions about what happens to a dependent variable (x) in the future • EX: businesses want to know what happens when they keep producing an item…will their initial investment will be worth it? How much money will they be making?

  7. EX 1: Real Life Applications of Limits You are manufacturing an iPhone case that costs $0.50 to produce. Your initial investment is $5000. • The cost of producing x cases is then given by C = • The average cost is then given by What is the average cost of producing 100 cases? 100,000 cases? What if you produce an infinite amount? • 0.50x+5000.

  8. Defined by Limits, What is a horizontal asymptote? The line y = b is a horizontal asymptote of the graph of a function y = f(x) if either When we say “The limit of f as x approaches infinity” we mean the limit of f as x moves increasingly far to the right on the number line.

  9. What algebraic rules AKA “shortcuts” can I use to help me to find limits at infinity? Consider the rational function f(x) = Then the limit as x approaches positive OR negative infinity is given by If m > n then the limit does not exist (it will be ±∞)

  10. EX 2A: Looking for Horizontal Asymptotes Given f(x) = what is:

  11. EX 2B: Looking for HA Given f(x) = , find: * Use graphing calc to find this

  12. You Try! Find and by looking at the graph and if applicable, with the rules . Identify all horizonal asymptotes. • f(x) = b) f(x) = c) f(x) = Treat |x| like a piecewise!

  13. Review from ALG 2: What is End Behavior?

  14. What is the end behavior model? The function g is a right end behavior model for f iff a left end behavior model for f iff

  15. EX 3A: Finding End Behavior Let f(x) = 3- 2 . We know that the end behavior model would be g(x) = 3. Show that while f and g are quite different for numerically small values of x, they are virtually identical for |x|. Zoom out

  16. You Try! Given f(x) = find the end behavior model. Then find and

  17. EX 3B Given f(x) = find the end behavior model. Then find and

  18. You Try! b) Find an end behavior model for f(x) = a) Find an end behavior model for f(x) =

  19. What are some other “special” limits at infinity? *memorize these* = 0 = DNE a) Find b)

  20. = 0 *Theorem* = = = = c) sin(1/x) Notice… sin(x) = 0

  21. Theorem 5: Properties of Limits as x→ Don’t need to copy down. Review on Pg 71 in textbook. If L, M, c and k are real numbers and and • Sum Rule: • The limit of the sum of two functions is the sum of their limits • +M • Difference Rule: • The limit of the difference of two function is the difference of the limits • M • The Product Rule: • The limit of a product of two functions is the product of their limits • M

  22. Cont. 4. Constant Multiple Rule: • The limit of a constant times a function is the constant times the limit of the function. 5. Quotient Rule: • The limit of a quotient of two functions is the quotient of their limits, provided the denominator is not zero. • = 6. Power Rule: • If r and s are integers, s ≠ 0, then

  23. EX 3: Finding HA Using Theorem 5 You Try! Find • Find • )

  24. Defined by Limits, What is a vertical asymptote? The line x = a is a vertical asymptote of the graph of a function y = f(x) if either

  25. EX 4: Finding Vertical Asymptotes Find the vertical asymptote(s) of each function. Describe the behavior to the left and right of each vertical asymptote with limits. a) f(x) = . a) f(x) = tan(x)

  26. *Note* You might think that the graph of a quotient always has a vertical asymptote where the denominator is zero, but this is not the case. There could be a hole. EX: for f(x) = = 1

  27. You Try! Find the vertical asymptotes of the graph of f(x) and then describe the behavior to the left and right of each vertical asymptote. a) g(x) = b) h(x) = c) f(x) = sec(x)

  28. “Four Corners” worksheet

  29. HW Pg 75 QR #7-9all EX 1-7odd, 13-17all (sketch all graphs with asymptotes clearly marked), 23, 27-31all, 33, 35-44all, 53-56all

  30. Review Activity

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