1. 2.2 Limits Involving Infinity Hoh Rainforest, Olympic National Park, WA

2. Limits & Infinity Consider What happens as x gets to be a big number? What happens as x gets even bigger?

3. Limits & Infinity Consider What if x is wicked huge? What if x equals infinity?

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

5. This number becomes insignificant as. There is a horizontal asymptote at 1. Horizontal Asymptotes

7. Find: When we graph this function, the limit appears to be zero. so for : by the sandwich theorem: Horizontal Asymptotes

8. Find: Horizontal Asymptotes

9. Limits & Infinity Consider What if x is wicked small? What if x is an infinitely small positive number? Whenever a denominator of a fraction approaches zero, the fraction equals +/ - infinity

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

11. The denominator is positive in both cases, so the limit is the same. Vertical Asymptotes: Is this really true?DNE

12. Find the Vertical Asymptotes: Vertical @ x= 2, 3 To find whether the asymptote goes up or down, you must examine the limits.

13. Find the Vertical Asymptotes: Vertical @ x= 2, 3 To find whether the asymptote goes up or down, you must examine the limits.

14. Find the Vertical & Horizontal Asymptotes:

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

17. Find an end behavior model for

18. End Behavior Models: Show that y = 3x 4 is an end behavior model for y = 3x 4 - 2x 2 – 7x +8.

19. End Behavior Models: Show that y = 2 is an end behavior model for

20. Test of model Our model is correct. End Behavior Models As, approaches zero. (The x term dominates.) becomes a right-end behavior model.becomes a left-end behavior model. As, increases faster than x decreases, therefore is dominant. Test of model Our model is correct.

21. becomes a right-end behavior model.becomes a left-end behavior model. On your calculator, graph: Use:

22. Right-end behavior models give us: dominant terms in numerator and denominator

23. Often you can just “think through” limits. 