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2.4:Introduction To Limits

2.4:Introduction To Limits. Objectives:. Understand the concept of a limit Calculate limits using graphs and tables. Greg Kelly, Hanford High School, Richland, Washington Modfied: Mike Efram Healdsburg HS 2004. How do Limits relate to the rest of Calculus?.

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2.4:Introduction To Limits

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  1. 2.4:Introduction To Limits Objectives: • Understand the concept of a limit • Calculate limits using graphs and tables Greg Kelly, Hanford High School, Richland, Washington Modfied: Mike Efram Healdsburg HS 2004

  2. How do Limits relate to the rest of Calculus? • The Tangent Problem – How can you find the equation of the line that is tangent to a given function at a given point? • The Area Problem – How can you find the area of a region between a graph of a function and the x-axis over some interval [a, b]? …we first must understand the concept of a Limit!!!

  3. WINDOW GRAPH Y= Consider: What happens as x approaches zero? Graphically:

  4. Looks like y=1

  5. TblSet TABLE Numerically: You can scroll down to see more values.

  6. It appears that the limit of as x approaches zero is 1 TABLE You can scroll down to see more values.

  7. “The limit of fof x as x approaches c from the right side is ” “The limit of fof x as x approaches c from the left side is ” One-sided Limits Limit notation: Limit notation:

  8. “The limit of fof x as x approaches c is L.” If: Then: Limit notation: Memorize So:

  9. The limit of a function refers to the value that the function approaches, not the actual value (if any). not 1

  10. Properties of Limits: Limits can be added, subtracted, multiplied, multiplied by a constant, divided, and raised to a power. (See page 119 for details.) For a limit to exist, the function must approach the same value from both sides. One-sided limits approach from either the left or right side only.

  11. does not exist because the left and right hand limits do not match! left hand limit right hand limit value of the function 2 1 1 2 3 4 At x=1:

  12. because the left and right hand limits match. left hand limit right hand limit value of the function 2 1 1 2 3 4 At x=2:

  13. because the left and right hand limits match. left hand limit right hand limit value of the function 2 1 1 2 3 4 At x=3:

  14. Common situations where a limit does not exist (DNE)!! • f(x) approaches a different number from the right side of c than the left side of c • f(x) increases or decreases without bound as x approaches c. Note: the notation is used to indicate the behavior of the function, even though the limit DNE. 3) f(x) oscillates between two fixed values as x approaches c.

  15. Show that: The maximum value of sine is 1, so The minimum value of sine is -1, so So: The Sandwich (or Squeeze) Theorem:

  16. Y= WINDOW By the sandwich theorem:

  17. p

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