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Welcome to Class 

Learn how to graph limits and evaluate them algebraically using techniques such as direct substitution, factoring, rationalizing roots, and multiplying by 1. Understand when limits do not exist and how to deal with undefined and indeterminate forms.

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Welcome to Class 

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  1. Welcome to Class  A quote for today: Success is the sum of small efforts, repeated day in and day out.Robert Collier (Inspirational Author) Today’s Warmup/Arrival Activity: Packet p. 1 & 2—“Pushed Beyond the Limit?” Answer: “There are no limits in life.”

  2. Draw a graph such that

  3. Draw a graph such that

  4. Draw a graph such that

  5. Two Cases for When the Limit is D.N.E. (Does Not Exist) • Behavior differs from the left and right • Oscillating Behavior • Ex/

  6. Let’s Practice Graphing Piecewise Functions Ourselves

  7. Example Problem #2

  8. Day 2 Evaluating Limits Algebraically

  9. Algebraic Techniques • Direct substitution • Factoring • Rationalizing roots • Multiply by 1 • Sometimes the limit does not exist

  10. Based on yesterday’s notes: However, for continuous functions Is often independent of f(c) We will refer to this as DIRECT SUBSTITUTION to find a limit.

  11. Example

  12. DIRECT SUBSTITUTION can be used if. . . • The function is continuous at c. • Which means (informally): No holes No jumps No vertical asymptotes

  13. Dealing with “holes” . . . . f(-4) D.N.E., however if we factor we can find

  14. Practice:

  15. Dealing with zero Undefined and indeterminate are NOT the same thing! Undefined is an answer and means it does not exist. Indeterminate means we do not know the answer…yet.

  16. “Indeterminate Form”: when direct substitution produces 0/0 Remember Conjugates?!?...

  17. You try:

  18. Back in Time

  19. Multiply by 1 in a “convenient form” (The common denominator)

  20. Summary • Direct substitution • Factoring • Rationalizing roots • Multiply by 1 • Sometimes the limit does not exist

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