1. 3.4 Review: Limits at Infinity Horizontal Asymptotes

2. 2 Limits at Infinity; Horizontal Asymptotes

3. 3 Example As x becomes arbitrarily large (positive or negative) what happens to y? Example: y = 1 is a Horizontal Asymptote

4. 4 More Examples:

5. 5 The curve y = f (x) has both y = –1 and y = 2 as horizontal asymptotes because and

6. 6 Practice 1 Evaluate - Find if there are any horizontal asymptotes.

7. 7 Practice 1 – Solution

8. 8 Practice 2 Find the horizontal and vertical asymptotes of the graph of the function Solution: Dividing both numerator and denominator by x and using the properties of limits, we have (since – x for x > 0)

9. 9 Practice 2 – Solution Therefore the line y = is a horizontal asymptote of the graph of f. cont’d

10. 10 Practice 2 – Solution In computing the limit as x  –, we must remember that for x < 0, we have = | x | = –x. So when we divide the numerator by x, for x < 0 we get cont’d

11. 11 Practice 2 – Solution Thus the line y = – is also a horizontal asymptote. A vertical asymptote is likely to occur when the denominator, 3x – 5, is 0, that is, when cont’d

12. 12 Infinite Limits at Infinity

13. 13 Infinite Limits at Infinity The notation is used to indicate that the values of f (x) become large as x becomes large. Similar meanings are attached to the following symbols:

14. 14 Example: Find and Solution: When becomes large, x 3 also becomes large. For instance, In fact, we can make x 3 as big as we like by taking x large enough. Therefore we can write

15. 15 Example – Solution Similarly, when x is large negative, so is x 3. Thus These limit statements can also be seen from the graph of y = x 3 in Figure 10. cont’d Figure 10