Limits and Derivatives
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Limits and Derivatives. 2. Limits Involving Infinity. 2.5. Limits Involving Infinity. In this section we investigate the global behavior of functions and, in particular, whether their graphs approach asymptotes, vertical or horizontal. Infinite Limits. Infinite Limits.

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Limits and Derivatives

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Limits and derivatives

Limits and Derivatives

2


Limits and derivatives

Limits Involving Infinity

2.5


Limits involving infinity

Limits Involving Infinity

  • In this section we investigate the global behavior of

  • functions and, in particular, whether their graphs approach

  • asymptotes, vertical or horizontal.


Limits and derivatives

Infinite Limits


Infinite limits

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


Infinite limits1

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.


Infinite limits2

Infinite Limits

  • Another notation for limxa f(x) = is

  • f(x)  as x  a


Infinite limits3

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


Infinite limits4

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


Infinite limits5

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


Infinite limits6

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.


Infinite limits7

Infinite Limits

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

Figure 4


Infinite limits8

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.


Example 1 evaluating one sided infinite limits

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


Example 1 solution

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


Example 1 solution1

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


Infinite limits9

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


Infinite limits10

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


Limits and derivatives

Limits at Infinity


Limits at infinity

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.


Limits at infinity1

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


Limits at infinity2

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


Limits at infinity3

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 


Limits at infinity4

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.


Limits at infinity5

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


Limits at infinity6

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


Limits at infinity7

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


Limits at infinity8

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.


Limits at infinity9

Limits at Infinity

  • For instance, the curve illustrated in Figure 8

  • has the line y = 1 as a horizontal asymptote because

Figure 8


Limits at infinity10

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


Example 3 infinite limits and asymptotes from a graph

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


Example 3 solution

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.


Example 3 solution1

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.


Limits at infinity11

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.


Limits at infinity12

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


Limits and derivatives

Infinite Limits at Infinity


Infinite limits at infinity

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:


Infinite limits at infinity1

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


Example 9 finding an infinite limit at infinity

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.


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