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2.2 Limits Involving Infinity

2.2 Limits Involving Infinity. The symbol . The symbol  means unbounded in the positive direction. (- in negative direction) It is NOT a number!. The line y = 0 is a horizontal asymptote for f.

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2.2 Limits Involving Infinity

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  1. 2.2 Limits Involving Infinity

  2. The symbol  • The symbol  means unbounded in the positive direction. (- in negative direction) • It is NOT a number!

  3. The line y = 0 is a horizontal asymptote for f As the denominator gets larger, the value of the fraction gets smaller. In other words as x gets larger positively or negatively, the y-values get closer to zero. The line y = b is a horizontal asymptote if: or

  4. vertical asymptote at x=0. As the denominator approaches zero from the left, the value of the fraction gets very large. As the denominator approaches zero from the right, the value of the fraction gets very large negatively.

  5. Review: Finding Asymptotes • 1st make sure R(x) = p(x)/q(x) is in simplest terms

  6. Examples: Find asymptotes and graph

  7. Vertical Asymptotes- Infinite Limits • The vertical line x = a is a vertical asymptote of a function y = f(x) if • If • If

  8. Graphically

  9. Examples: Find the limits graphically and numerically

  10. Examples: Find the limits graphically and numerically

  11. Horizontal Asymptotes – Limits at Infinity • The line y = b is a horizontal asymptote of y = f(x) if either The limit at infinity is also referred to as end behavior.

  12. Examples: Find the limits at infinity graphically and numerically

  13. Finding the limit at infinity analytically • If f(x) is a rational function then to find the limit at infinity simply find the horizontal asymptote using the rules about degrees.

  14. Examples

  15. Theorem

  16. Non-rational functions • If the function is not a rational function then you can try: • Dividing top and bottom by highest power on bottom • Rationalizing • Rewriting the problem

  17. Examples: Divide

  18. Example: Rationalize

  19. Example: Rewrite

  20. End Behavior Models • Graph on the window [-20, 20] by [-1000000, 5000000] Notice as the graphs become identical. We say that g(x) act as a model for f(x) as or g(x) is an end behavior model for f(x)

  21. Example • Show graphically that g(x) = x is a right end behavior model and h(x) = e-x is a left end behavior model for f(x) = x + e-x

  22. End behavior models for polynomials • If

  23. Examples: Find the end behavior model

  24. HW: p. 71 • 1-22,29-38 • Worksheet

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