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Notes 2.1 - Limits

Notes 2.1 - Limits. I. Limit Notation. A.) Two-sided Notation: Read as “the limit of f ( x ) as x approaches a is L .”. B.) One-sided Notation: Read as “the limit of f ( x ) as x approaches a from the right is L .”

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Notes 2.1 - Limits

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  1. Notes 2.1 - Limits

  2. I. Limit Notation A.) Two-sided Notation: Read as “the limit of f(x) as x approaches a is L.”

  3. B.) One-sided Notation: Read as “the limit of f(x) as x approaches a from the right is L.” Read as “the limit of f(x) as x approaches a from the left is L.”

  4. II. Limit Definition A.) Def: The function f has a limit L as x approaches ciff: B.) Visual Representations:

  5. B.) Visual Reps(cont.)-

  6. III. Non-existent Limits A.) fails to existwhen: 1.) The right-side limit and left-side limit equal different real numbers. 2.) The are infinite oscillations . 3.) The limit(s) approach

  7. Ex. – Evaluate Although limits approaching infinity do not exist, we must still describe the behavior from both/each side(s)!!!

  8. IV. Computational Techniques for Limits A.) Properties of Limits: p. 61-62 of your text. KNOW THEM!!! SUM, DIFFERENCE, PRODUCT, CONSTANT MULTIPLE, QUOTIENT, and POWER RULES

  9. B.) Theorem – Polynomial and Rational Functions-

  10. B.) Theorem – Polynomial and Rational Functions-

  11. C.) Limit Properties and Theorems valid for one-sided limits as well.

  12. V. Techniques for Evaluation A.) Apply the theorems/properties (Substitute) B.) Modify the expression: 1.) Simplify 2.) Factor 3.) Expand 4.) Multiply by conjugates 5.) Apply thm./prop. again

  13. C.) Evaluate the following limits: 1.) 2.)

  14. 3.) 4.)

  15. 5.)

  16. 6.)

  17. 7.)

  18. 8.)

  19. 9.)

  20. VI. You Try! 1.) 2.) 3.) 4.)

  21. VII. Sandwich Theorem GRAPHICALLY

  22. B.) Example - What do you know about the sin function?

  23. C.) Example -

  24. VIII. Limit Theorems

  25. IX. Proof of Given the point P in the first quadrant on the unit circle: The area of sector OAP = P (x, y) 1 O (0, 0) 𝜃 A (1, 0)

  26. Dropping a perp. From P to Q on the x-axis gives us triangle OPQ The area of triangle OPQ = P (x, y) 1 𝜃 O (0, 0) Q (x,0) A (1, 0)

  27. Now, we can all agree that P (x, y) 1 𝜃 O (0, 0) Q (x,0) A (1, 0)

  28. Extending OP to T and dropping a perp. from T to A on the x-axis gives us triangle OTA The area of triangle OTA = T(1, tan 𝜃) P 1 𝜃 O (0, 0) A Q

  29. Now, we can all agree that T(1, tan 𝜃) P 1 𝜃 O (0, 0) A Q

  30. X. Patching In order to make our trigonometric limits look like A-D of II, we may need to “PATCH” the trig expression. After, we apply our limit properties and verify on our calculator. A) Examples -

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