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Mastering Limits and Asymptotes in AP Calculus AB/BC

Learn about limits involving infinity and horizontal/vertical asymptotes, with examples and properties to help you understand calculus concepts better. Use graphs and tables to identify asymptotes. Practice finding and describing vertical asymptotes in functions. Understand end behavior models. Enhance your AP Calculus skills today!

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Mastering Limits and Asymptotes in 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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