Limits Involving Infinity

1. Limits InvolvingInfinity 2.2

2. Finite limits as x→±∞ The symbol for infinity (∞) does not represent a real number. We use ∞ to describe the behavior of a function when the values in its domain or range outgrow all finite bounds. For example, when we say “the limit of f as x approaches infinity”we mean the limit of f as x moves increasingly far to the right on the number line. When we say “the limit of f as x approaches negative infinity (- ∞)”we mean the limit of f as x moves increasingly far to the left on the number line.

3. Horizontal Asymptotes

4. Example Horizontal Asymptote [-6,6] by [-5,5]

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

6. Find: When we graph this function, the limit appears to be zero. so for : by the sandwich theorem: Example 2:

7. Example 3: Find:

8. Properties of limits ¥

9. Infinite Limits as x→a

10. Vertical Asymptote

11. 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.

12. Vertical Asymptote

13. Example 4: The denominator is positive in both cases, so the limit is the same.

14. Example Vertical Asymptote [-6,6] by [-6,6]

15. End Behavior Models

16. End Point Behavior

17. As , approaches zero. becomes a right-end behavior model. As , increases faster than x decreases, therefore is dominant. becomes a left-end behavior model. Example 7: (The x term dominates.) Test of model Our model is correct. Our model is correct. Test of model

18. becomes a right-end behavior model. becomes a left-end behavior model. Example 7:

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

20. Often you can just “think through” limits. p