1. Limits Involving Infinity Infinite Limits We have concluded that

2. Limits Involving Infinity x1/x 2 ±11 ±0.54 ±0.225 ±0.1100 ±0.05400 ±0.0110,000 ±0.0011,000,000 By observing from the table of values and the graph, the values of 1/x 2 can be made arbitrarily large by taking x to be close enough to 0. The values of f(x) do not approach a number.

3. Limits Involving Infinity To indicate this kind of behavior, we use the notation This does not mean that is a number. Nor does it mean that the limit exists. It expresses the particular way the limit does not exist.

4. Limits Involving Infinity In general we write to indicate that the values of f(x) become larger and larger (or “increase without bound”) as x approaches a. Graphical illustration on page 131, figure 2 and figure 3 and figure 4

5. Limits Involving Infinity Definition: The notation means that the values of f(x) can be made arbitrarily large (as large as we please) by taking x sufficiently close to a (on either side of a) but not equal to a. This notation is often read as: “the limit of f(x), as x approaches a, is infinity” “f(x) becomes infinite as x approaches a” “f(x) increases without bound as x approaches a.

6. Limits Involving Infinity Similarly means that values of f(x) are as large negative as we like for all values of x that are sufficiently close to a, but not equal to a. Look at figure 4 on page 132.

7. Limits Involving Infinity Definition: The line x = a is called a vertical asymptote of the curve y = f(x) if at least one of the following statements is true:

8. Limits Involving Infinity For example, the y-axis is a vertical asymptote of the curve y = 1/x 2 because

9. Limits Involving Infinity Example: Find and. Look at the graph of this function on your graphing calculator!

10. Limits Involving Infinity and

11. Limits Involving Infinity Example: Find.

12. Limits Involving Infinity Limits at Infinity Let’s look at the graph of As you can see from the graph, as x grows larger and larger, the values of f(x) get closer and closer to 1.

13. Limits Involving Infinity This is expressed symbolically by writing

14. Limits Involving Infinity Definition: Let f be a function defined on some interval (a, ). Then means that the values of f(x) can be made as close to L as we like by taking x sufficiently large.

15. Limits Involving Infinity This is often read as: “the limit of f(x), as x approaches infinity, is L” “the limit of f(x), as x becomes infinite, is L” “the limit of f(x), as x increases without bound, is L”

16. Limits Involving Infinity Look at the illustrations on page 134, figure 9.

17. Limits Involving Infinity Definition: The line y = L is called a horizontal asymptote of the curve y = f(x) if either or Therefore the curve has a horizontal asymptote at y = 1.

18. Limits Involving Infinity An example of a curve with two horizontal asymptotes is y = tan -1 x.

19. Limits Involving Infinity Find the infinite limits, limits at infinity, and asymptotes for the function f whose graph is shown in figure 12 (page 135)

20. Limits Involving Infinity Example: Find and We now have proven that y = 0 is a horizontal asymptote for the graph of y = 1/x.

21. Limits Involving Infinity Important Rule for Calculating Limits If n is a positive integer, then

22. Limits Involving Infinity Example: Evaluate To evaluate the limit at infinity of a rational function, we first divide both the numerator and denominator by the highest power of x. (We assume that x does not equal zero, since we are interested only in large values of x)

23. Limits Involving Infinity Example: Compute

24. Limits Involving Infinity

25. Example: Evaluate

26. Limits Involving Infinity Example: Evaluate