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Understanding Limits: Properties and Evaluating Different Forms

This introduction explores the essential concept of limits in calculus. Learn how to evaluate limits by plugging in values to functions and understand different forms of limits, including 'easy limits' and '0/0' indeterminate forms. The guide emphasizes that it is what x approaches that matters, not the actual value of x. Through varied examples, including the slope of curves using the difference quotient, students will gain practical skills in determining limits effectively.

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Understanding Limits: Properties and Evaluating Different Forms

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  1. PAGE 917 14.1: Introduction to Limits

  2. PROPERTIES OF LIMITS SECTION 14.2

  3. DIFFERENT FORMS OF LIMITS • Always plug in the number to confirm the form of limit • It will fit into these forms of limits: 1. Easy Limits: Plug in x to f(x) to get the limit 2. “0/0” Limits • Plug in x to prove 0/0 • Apply Algebra Factoring Rules • Plug in x to f(x) to get the limit REMEMBER: IT IS WHAT X APPROACHES NOT WHAT X IS 14.1: Introduction to Limits

  4. EXAMPLE 1 Evaluate 14.1: Introduction to Limits

  5. EXAMPLE 2 Evaluate 14.1: Introduction to Limits

  6. EXAMPLE 3 Evaluate 14.1: Introduction to Limits

  7. EXAMPLE 4 Evaluate 14.1: Introduction to Limits

  8. YOUR TURN Evaluate 14.1: Introduction to Limits

  9. EXAMPLE 5 Use given below, determine 14.1: Introduction to Limits

  10. EXAMPLE 6 Use given below, determine 14.1: Introduction to Limits

  11. YOUR TURN Use given below, determine 14.1: Introduction to Limits

  12. EXAMPLE 7 Evaluate What form is this? AS X APPROACHES 4, F(X) OR Y APPROACHES 8. 14.1: Introduction to Limits

  13. EXAMPLE 7 Evaluate 14.1: Introduction to Limits

  14. EXAMPLE 8 Evaluate 14.1: Introduction to Limits

  15. YOUR TURN Evaluate 14.1: Introduction to Limits

  16. EXAMPLE 9 Evaluate What form is this? NO NEED TO FOIL THE BOTTOM 14.1: Introduction to Limits

  17. EXAMPLE 9 Evaluate 14.1: Introduction to Limits

  18. EXAMPLE 10 Evaluate 14.1: Introduction to Limits

  19. YOUR TURN Evaluate 14.1: Introduction to Limits

  20. DIFFERENCE QUOTIENT • The Limit of a Difference Quotient is used to find the slope of the curves • It is also the slope of the tangent line 14.1: Introduction to Limits

  21. EXAMPLE 11 Given f(x) = 2x2, find the slope at x = –3 14.1: Introduction to Limits

  22. EXAMPLE 11 Given f(x) = 2x2, find the slope at x = –3 14.1: Introduction to Limits

  23. EXAMPLE 12 Given f(x) = x2, find the slope at x = 5 14.1: Introduction to Limits

  24. YOUR TURN Given f(x) = x2 – 1, find the slope at x = 3 14.1: Introduction to Limits

  25. ASSIGNMENT Worksheet 14.1: Introduction to Limits

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