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Limits and Derivatives

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Limits and Derivatives

2

Limits Involving Infinity

2.5

- In this section we investigate the global behavior of
- functions and, in particular, whether their graphs approach
- asymptotes, vertical or horizontal.

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

- Thus the values of f(x) do not approach a number, so
- limx0 (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.

- Another notation for limxa f(x) = is
- f(x) as x a

- Again, the symbol is not a number, but the expression
- limxa 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

- 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

- The symbol limxa 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

- 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 “xa+”means that we consider only x > a.

- Illustrations of these four cases are given in Figure 4.

Figure 4

- For instance, the y-axis is a vertical asymptote of the curve
- y = 1/x2 because limx0 (1/x2) = .
- In Figure 4 the line x = a is a vertical asymptote in each of the four cases shown.

- 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

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

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

- 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

- 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

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.

- 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

- 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

- 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 limxf(x) = L is
- f(x) L as x

- 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.

Examples illustrating

- 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

- 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

Examples illustrating

- 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

- 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.

- For instance, the curve illustrated in Figure 8
- has the line y = 1 as a horizontal asymptote because

Figure 8

- The curve y = f(x) sketched in Figure 11 has both y = –1 and y = 2 as horizontal asymptotes because

Figure 11

- Find the infinite limits, limits at infinity, and asymptotes for the function f whose graph is shown in Figure 12.

Figure 12

- 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.

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.

- 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.

- 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

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:

- 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

- 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.