Limits and Derivatives

1. Limits and Derivatives 2

2. Limits Involving Infinity 2.5

3. Limits Involving Infinity • In this section we investigate the global behavior of • functions and, in particular, whether their graphs approach • asymptotes, vertical or horizontal.

4. Infinite Limits

5. Infinite Limits • We have concluded that • does not exist • by observing, from the table of values and the graph of y = 1/x2 in Figure 1, that the values of 1/x2 can be made arbitrarily large by taking x close enough to 0. Figure 1

6. Infinite Limits • Thus the values of f(x) do not approach a number, so • limx0 (1/x2) does not exist. • To indicate this kind of behavior we use the notation • In general, we write symbolically • to indicate that the values of f(x) become larger and larger (or “increase without bound”) as x approaches a.

7. Infinite Limits • Another notation for limxa f(x) = is • f(x)  as x  a

8. Infinite Limits • Again, the symbol is not a number, but the expression • limxa f(x) = is often read as • “the limit of f(x), as x approaches a, is infinity” • or “f(x) becomes infinite as x approaches a” • or “f(x) increases without bound as x approaches a” • This definition is illustrated • graphically in Figure 2. Figure 2

9. Infinite Limits • Similarly, as shown in Figure 3, • means that the 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. Figure 3

10. Infinite Limits • The symbol limxa f(x) = can be read as “the limit of f(x), • as x approaches a, is negative infinity” or “f(x) decreases • without bound as x approaches a.” • As an example we have

11. Infinite Limits • Similar definitions can be given for the one-sided infinite limits • remembering that “x a–”means that we consider only values of x that are less than a, and similarly “xa+”means that we consider only x > a.

12. Infinite Limits • Illustrations of these four cases are given in Figure 4. Figure 4

13. Infinite Limits • For instance, the y-axis is a vertical asymptote of the curve • y = 1/x2 because limx0 (1/x2) = . • In Figure 4 the line x = a is a vertical asymptote in each of the four cases shown.

14. Example 1 – Evaluating One-sided Infinite Limits • Find and • Solution: • If x is close to 3 but larger than 3, then the denominator • x – 3 is a small positive number and 2x is close to 6. • So the quotient 2x/(x – 3) is a large positive number. • Thus, intuitively, we see that

15. Example 1 – Solution cont’d • Likewise, if x is close to 3 but smaller than 3, then x – 3 is a small negative number but 2x is still a positive number • (close to 6). • So 2x/(x – 3) is a numerically large negative number. • Thus

16. Example 1 – Solution cont’d • The graph of the curve y = 2x/(x – 3) is given in Figure 5. • The line x = 3 is a vertical asymptote. Figure 5

17. Infinite Limits • Two familiar functions whose graphs have vertical asymptotes are y = ln x and y = tan x. • From Figure 6 we see that • and so the line x = 0 (the y-axis) • is a vertical asymptote. • In fact, the same is true for • y = loga x provided that a > 1. Figure 6

18. Infinite Limits • Figure 7 shows that • and so the line x = /2 is a vertical asymptote. • In fact, the lines x = (2n + 1)/2, • n an integer, are all vertical • asymptotes of y = tan x. y = tan x Figure 7

19. Limits at Infinity

20. Limits at Infinity • In computing infinite limits, we let x approach a number and the result was that the values of y became arbitrarily large (positive or negative). • Here we let x become arbitrarily large (positive or negative) and see what happens to y. • Let’s begin by investigating the behavior of the function f defined by • as x becomes large.

21. Limits at Infinity • The table at the right gives values • of this function correct to six decimal • places, and the graph of f has been • drawn by a computer in Figure 8. Figure 8

22. Limits at Infinity • As x grows larger and larger you can see that the values of • f(x) get closer and closer to 1. • In fact, it seems that we can make the values of f(x) as close as we like to 1 by taking x sufficiently large. • This situation is expressed symbolically by writing

23. Limits at Infinity • In general, we use the notation • to indicate that the values of f(x) approach L as x becomes larger and larger. • Another notation for limxf(x) = L is • f(x) L as x 

24. Limits at Infinity • The symbol does not represent a number. • Nonetheless, the expression is often read as • “the limit of f(x), as x approaches infinity, is L” • or “the limit of f(x), as x becomes infinite, is L” • or “the limit of f(x), as x increases without bound, is L” • The meaning of such phrases is given by Definition 4.

25. Examples illustrating Limits at Infinity • Geometric illustrations of Definition 4 are shown in Figure 9. • Notice that there are many ways for the graph of f to approach the line y = L (which is called a horizontal asymptote) as we look to the far right of each graph. Figure 9

26. Limits at Infinity • Referring to Figure 8, we see that for numerically large negative values of x, the values of f(x) are close to 1. • By letting x decrease through negative values without bound, we can make f(x) as close to 1 as we like. • This is expressed by writing Figure 8

27. Examples illustrating Limits at Infinity • In general, as shown in Figure 10, the notation • means that the values of f(x) can be made arbitrarily close to • L by taking x sufficiently large negative. Figure 10

28. Limits at Infinity • Again, the symbol does not represent a number, but the • expression is often read as • “the limit of f(x), as x approaches negative infinity, is L” • Notice in Figure 10 that the graph approaches the line y = L • as we look to the far left of each graph.

29. Limits at Infinity • For instance, the curve illustrated in Figure 8 • has the line y = 1 as a horizontal asymptote because Figure 8

30. Limits at Infinity • The curve y = f(x) sketched in Figure 11 has both y = –1 and y = 2 as horizontal asymptotes because Figure 11

31. Example 3 – Infinite Limits and Asymptotes from a Graph • Find the infinite limits, limits at infinity, and asymptotes for the function f whose graph is shown in Figure 12. Figure 12

32. Example 3 – Solution • We see that the values of f(x) become large as x  –1 from both sides, so • Notice that f(x) becomes large negative as x approaches 2 from the left, but large positive as x approaches 2 from the right. So • Thus both of the lines x = –1 and x = 2 are vertical asymptotes.

33. Example 3 – Solution cont’d • As x becomes large, it appears that f(x) approaches 4. • But as x decreases through negative values, f(x) approaches 2. So • This means that bothy = 4 and y = 2 are horizontal asymptotes.

34. Limits at Infinity • Most of the Limit Laws hold for limits at infinity. It can be • proved that the Limit Laws are also valid if “x  a” is replaced by “x  ” or “x  .” • In particular, we obtain the following important rule for • calculating limits.

35. Limits at Infinity • The graph of the natural exponential function y = exhas the line y = 0 (the x-axis) as a horizontal asymptote. (The same is true of any exponential function with base a > 1.) • In fact, from the graph in Figure 16 and the corresponding table of values, we see that • Notice that the values of • ex approach 0 very rapidly. Figure 16

36. Infinite Limits at Infinity

37. 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:

38. Infinite Limits at Infinity • From Figures 16 and 17 we see that • but, as Figure 18 demonstrates, y = ex becomes large as • x at a much faster rate than y = x3. Figure 17 Figure 18 Figure 16

39. Example 9 – Finding an Infinite Limit at Infinity • Find • Solution: • It would be wrong to write • The Limit Laws can’t be applied to infinite limits because • is not a number ( can’t be defined). • However, we can write • because both x and x – 1 become arbitrarily large.