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Mastering Limit Evaluation: Techniques and Examples

Learn to evaluate limits algebraically and graphically, starting with direct substitution and progressing to advanced methods like the Sandwich Theorem. Discover ways to simplify expressions with factors and conjugates for accurate substitution. Through examples, understand when direct substitution fails and how to employ clever techniques for successful limit evaluation. Improve your understanding of calculus with clear guidance on expanding and reducing fractions effectively.

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Mastering Limit Evaluation: Techniques and Examples

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  1. 1.03 Evaluating Limits Algebraically

  2. Graphically (√) Algebraically (starting this lesson) Using the Sandwich Theorem (later) Evaluating Limits

  3. Direct substitution (easy!) Expand, simplify  substitution Factor, simplify by reducing  substitution Combine to a single fraction, reduce  substitute Simplify using multiplication by conjugate Simplify using “known” limits Ways to algebraically evaluate

  4. Example 1 • Try direct substitution

  5. Example 2 • Try direct substitution • (remember, we use RADIANS in calculus)

  6. Example 3 • Direct substitution clearly gives us division by 0 • LIMIT IS NOT 0 !!!!

  7. Example 3 • Expand and simplify • Notice that the “lim” stays in the work until you substitute

  8. Example 4 • Try direct substitution • Doesn’t work… division by 0!

  9. Example 4 • Factor and reduce

  10. Example 5 • Factor and reduce

  11. Example 6 • Check – does direct substitution work? • Division by zero • No common factors?

  12. Example 6 • Use the “conjugate trick” • (or be clever and factor denominator)

  13. Always perform a mental check to see if direct substitution will work right off the top… very ugly looking limits can actually be really easy! Pay careful attention to how you expand and how you reduce fractions Watch out

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