1.5 Infinite limits

1. 1.5 Infinite limits "I never got a pass mark in math ... Just imagine -- mathematicians now use my prints to illustrate their books." -- M.C. Escher

2. Objective: • To describe infinite limits

3. Black holes • Start with any number • Count the number of even digits, the number of odd digits, the total number of digits. • Write that 3-digit number • Repeat • Repeat • Repeat

4. Ways limits DNE • Limit from the left is different than the limit from the right • Function increases or decreases without bound • Function oscillates

5. Function increases or decreases without bound • If both the left and the right side approach infinity then • If both the left and the right side approach negative infinity then

7. Discontinuities • 2 types: • Removable • Non-removable

8. Def. of a vertical asymptote • If f(x) approaches infinity or negative infinity as x approaches c from the right or left then the line x = c is a v. a. of the graph

9. V. A. theorem • The functions f and g are continuous on an open interval. If f(c) does not equal zero, g(c) = 0, and g(x) is not zero for all other x in the interval then has a v. a. at x = c

10. In other words • Look for zeros in the denominator and then check the numerator to see if it is a hole or an asymptote

11. Examples

12. More examples

13. Limits and V. A. • Find: • What do you know about the function?

14. Left and right limits

15. Cont.. • Check from the left: • Check from the right: • The limit is…

16. Properties of limits • 1. Sum or difference: • 2. Product:

17. More properties • 3. Quotient • These are also true for negative infinity

18. Practice problems