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Understanding Trigonometric Limits: Properties, Theorems, and Evaluations

Delve into the properties of limits involving trigonometric functions in this comprehensive guide. Learn how to evaluate limits that present the indeterminate form "0/0" through techniques like direct substitution, factoring, simplification, and rationalization. Explore essential concepts such as the Squeeze Theorem, which allows for evaluating limits of bounded functions. Included are various examples, practice problems, and tips on recognizing key trigonometric limits, including why limits like sin(x)/x approaching 0 equals 1. Perfect for students looking to master trigonometric limits!

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Understanding Trigonometric Limits: Properties, Theorems, and Evaluations

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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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