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AP Calculus AB/BC

AP Calculus AB/BC. 2.2 Limits Involving Infinity, p. 70. As the denominator gets larger, the value of the fraction gets smaller. There is a horizontal asymptote if:. or. Definition: Horizontal Asymptote.

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AP Calculus AB/BC

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  1. AP Calculus AB/BC 2.2 Limits Involving Infinity, p. 70

  2. As the denominator gets larger, the value of the fraction gets smaller. There is a horizontal asymptote if: or

  3. Definition: Horizontal Asymptote • The line y = b is a a horizontal asymptote of the graph of a function y = f(x) if either: • For example: y = -3 is the horizontal asymptote.

  4. This number becomes insignificant as . There is a horizontal asymptote at 1. Example 1a

  5. Example 1b = 0 Use graphs and tables to find: • Identify all horizontal asymptotes. = -∞ y = 0 (the x-axis)

  6. Example 2 = 0 Use graphs and tables to find: • Identify all horizontal asymptotes. = 0 y = 0 First, graph on your calculator. Next, use the table feature on your calculator to look at extremely large and small values for x.

  7. by the sandwich theorem: Example 2 (cont.) Since then

  8. Theorem 5: Properties of Limits • Limits at infinity have properties similar to those of finite limits.

  9. Find: Example 3 pDay 1

  10. Infinite Limits: As the denominator approaches zero, the value of the fraction gets very large. vertical asymptote at x=0. If the denominator is positive then the fraction is positive. If the denominator is negative then the fraction is negative.

  11. Finding Vertical Asymptotes • The denominator of any function cannot equal zero. • So, set the denominator equal to zero and solve. • The result is/are a vertical asymptote.

  12. Example 4 • Find the vertical asymptotes of the graph of f(x). • Describe the behavior of f(x) to the left and right of each vertical asymptote. • Set x + 1 = 0. • The vertical asymptote is the line x = -1.

  13. Example 4 (cont.) • To describe the behavior of f(x), substitute values to the left and right of the vertical asymptote. • So, let x = −0.5, then f(x) is a positive number, so • Next, let x = −1.5, then f(x) is a negative number, so

  14. Example 7: Right-end behavior models give us: dominant terms in numerator and denominator

  15. End Behavior Models: Thefunction g is: a right end behavior model for f if and only if a left end behavior model for f if and only if End behavior models model the behavior of a function as x approaches infinity or negative infinity.

  16. As , approaches zero. becomes a right-end behavior model. becomes a left-end behavior model. Example 8: (The x term dominates.) Test of model Our model is correct. As , increases faster than x decreases, therefore is dominant. Test of model Our model is correct.

  17. becomes a right-end behavior model. becomes a left-end behavior model. Use: Example 8 (cont.): On your calculator, graph: p

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