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Page 54. Page 67. Evaluating Trig Limits with Trig Functions. Section 1.3. “0/0” Limits AKA: Indeterminate Form. Always begin with direct substitution Completely factor the problem

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Page 54

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  1. Page 54 1.3 - Properties of Limits with Trig

  2. Page 67 1.3 - Properties of Limits with Trig

  3. Evaluating Trig Limits with Trig Functions Section 1.3 1.3 - Properties of Limits with Trig

  4. “0/0” LimitsAKA: Indeterminate Form • Always begin with direct substitution • Completely factor the problem • Simplify and/or Cancel by identifying a function g that agrees with for all x except x = c. Take the limit of g • Apply algebra rules • If necessary, Rationalize the numerator • Plug in x of the function to get the limit 1.3 - Properties of Limits with Trig

  5. Example 1 • Solve What form is this? 1.3 - Properties of Limits with Trig

  6. Example 1 • Solve AS X APPROACHES 4, f(x) OR Y APPROACHES 8. 1.3 - Properties of Limits with Trig

  7. Example 1 (Calculator) • Solve 1.3 - Properties of Limits with Trig

  8. Example 2 • Solve 1.3 - Properties of Limits with Trig

  9. Your Turn • Solve 1.3 - Properties of Limits with Trig

  10. When in Algebra… • You learned to: NO RADICALS IN THE DENOMINATOR IN LIMITS, NO RADICALS IN THE NUMERATOR and DENOMINATOR 1.3 - Properties of Limits with Trig

  11. Example 3 • Solve What form is this? 1.3 - Properties of Limits with Trig

  12. Example 3 • Solve NO NEED TO FOIL THE BOTTOM 1.3 - Properties of Limits with Trig

  13. Example 3 • Solve 1.3 - Properties of Limits with Trig

  14. Example 4 • Solve 1.3 - Properties of Limits with Trig

  15. Your Turn • Solve Hint: Don’t combine like terms to the denominator, too early 1.3 - Properties of Limits with Trig

  16. Example 5 • Solve What form is this? 1.3 - Properties of Limits with Trig

  17. Example 5 • Solve 1.3 - Properties of Limits with Trig

  18. Example 5 • Solve 1.3 - Properties of Limits with Trig

  19. Example 5 • Solve 1.3 - Properties of Limits with Trig

  20. Example 6 • Solve 1.3 - Properties of Limits with Trig

  21. Your Turn • Solve 1.3 - Properties of Limits with Trig

  22. “Squeeze Theorem” • Also known as the “Sandwich theorem,” it is used to evaluate the limit of a function that can't be computed at a given point. • For a given interval containing point c, where f, g, and h are three functions that are differentiable and f(x) < g(x) < h(x) over the interval where f(x) is the upper bound and h(x) is the lower bound 1.3 - Properties of Limits with Trig

  23. “Squeeze Theorem” 1.3 - Properties of Limits with Trig

  24. Example 7 • Use the Squeeze Theorem to evaluate where c = 1 for 3x<g(x) <x3 + 2 1.3 - Properties of Limits with Trig

  25. Example 7 1.3 - Properties of Limits with Trig

  26. Example 8 • Use the Squeeze Theorem to evaluate for 4x – 9 <f(x) <x2 – 4x + 7 for which x> 0 1.3 - Properties of Limits with Trig

  27. Your Turn • Use the Squeeze Theorem to evaluate where c = 0 for 9 – x2<g(x) < 9 + x2 1.3 - Properties of Limits with Trig

  28. Two Special Trigonometric Limits • When expressing x in radians and not in degrees • The use help explains the “Squeeze” Theorem 1.3 - Properties of Limits with Trig

  29. Why is the limit of sin (x)/x, when x approaches 0 equal to 1? MEMORIZE IT! 1.3 - Properties of Limits with Trig

  30. Why is the limit of 1 – cos (x)/x, when x approaches 0 equal to 0? MEMORIZE IT! 1.3 - Properties of Limits with Trig

  31. Example 9 Is there another way of rewriting tan (x)? • Solve Split the fraction up so we can isolate and utilize a trigonometric limit 1.3 - Properties of Limits with Trig

  32. Example 9 • Solve Utilize the Product Property of Limits 1.3 - Properties of Limits with Trig

  33. Example 10 Try to convert it to one of its trig limits. • Solve Try to get it where the sine trig function to cancel. Whatever is applied to the bottom, must be applied to the top. 1.3 - Properties of Limits with Trig

  34. Example 10 • Solve 1.3 - Properties of Limits with Trig

  35. Example 11 • Solve 1.3 - Properties of Limits with Trig

  36. Your Turn • Solve 1.3 - Properties of Limits with Trig

  37. Pattern? • Solve = 4 • Solve = • Solve = • Solve = • Solve = 1.3 - Properties of Limits with Trig

  38. Example 12 Split the fraction up so we can isolate and utilize a trigonometric limit • Solve 1.3 - Properties of Limits with Trig

  39. Example 13 • Solve cos(0) = 1 1.3 - Properties of Limits with Trig

  40. Example 14 • Solve 1.3 - Properties of Limits with Trig

  41. Your Turn • Solve 1.3 - Properties of Limits with Trig

  42. AP Multiple Choice Practice Question (non-calculator) • Solve • π • 1 • 0 • –1 • Does Not Exist 1.3 - Properties of Limits with Trig

  43. AP Multiple Choice Practice Question (non-calculator) • Solve • [A] 0 • [B] –π/2 • [C] (2√2)/π • [D] 2/π • [E] None of these 1.3 - Properties of Limits with Trig

  44. AP Free Response Practice Question (non-calculator) • If a ≠ 0, then determine . If the limit does not exist, explain why. 1.3 - Properties of Limits with Trig

  45. Assignment • Page 67 • 67-77 all, 87 1.3 - Properties of Limits with Trig

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