1.06 Limits involving infinity

1. 1.06 Limits involving infinity

2. Limit  the expected / intended value of a function • A limit can involve ∞ in two ways: • You can expect a limit to be equal to ±∞ (vertical asymptote, limit DNE) • You can expect a value of a function as x approaches ±∞ (horizontal asymptote) Limits involving infinity

3. As x approaches 3, the value of this function approaches ∞… the limit does not exist Example 1

4. Example 2 From right RHL ≠ LHL; neither limit has a defined value From left

5. When you cannot “get rid of” division by zero in a limit, this represents a vertical asymptote The limit approaches ±∞ The limit does not exist Example 3

6. As x gets larger and larger (approaches ∞) what is the intended value of this function? Limits at infinity – example 4

7. If no horizontal asymptote exists Example 5

8. Example 6

9. Example 6

10. Example 6

11. To evaluate without graphing, we have to examine the simple limit, Using a calculator we can quickly determine that Asymptotes without a graph

12. In fact, through some investigating, we can determine that for any constant c and any power n>1 Asymptotes without a graph

13. To evaluate without graphing, you perform a factoring technique to reduce each term to lowest terms. Asymptotes without a graph

14. Look for the highest power of x and divide each term by this power. Asymptotes without a graph

15. Then reduce Asymptotes without a graph

16. Now we can use the fact that to remove/simplify our example Asymptotes without a graph

17. Asymptotes without a graph

18. Eg.