第5.8节 极限的（ε，δ）条件

5.8 THE ε, δ CONDITION FOR LIMITS

The traditional calculus course is developed entirely without infinitesimals. The starting point is the concept of a limit. The intuitive idea of ____________ is: For every real number x which is close to but not equal to c, f(x) is close to L.

It is hard to make this idea into a rigorous definition, because one must clarify the word “close”. Indeed, the whole point of our infinitesimal approach to calculus is that it is easier to define and explain limits using infinitesimals. The definition of limits in terms of real numbers is traditionally expressed using the Greek letters ε (epsilon) and δ (delta), and is therefore called the ε,δ condition for limits.

The ε, δ condition will be based on the notion of distance between two real numbers.

DEFINITION

The distance between two real numbers x and c is the absolute value of their difference,

distance = |x-c|.

x is within δ of c if |x-c| ≤ δ.

x is strictly within δ of c if |x-c| < δ.

Notice that the distance |x-c| is just the difference between the larger and the smaller of the two numbers x and c. This is a place where the absolute value sign is especially convenient. The following simple but helpful lemma is illustrated in Figure 5.8.1.

Figure 5.8.1

LEMMA

(i) x is within δ of c if and only if

c - δ ≤ x ≤ c +δ.

(ii) x is strictly within δ of c if and only if

c - δ < x < c +δ.

PROOF (i) Subtracting c from each term we see that

c - δ ≤ x ≤ c +δ.

if and only if - δ ≤ x - c ≤ δ,

Which is true if and only if |x-c| ≤ δ.

The proof of (ii) is similar.

We shall repeat our infinitesimal definition of limit from Section 3.3 and then write down the ε, δ condition for limits. Later we shall prove that the two definitions of limit are equivalent to each other.

Suppose the real function f is defined for all real numbers x ≠ c in some neighborhood of c.

DEFINITION OF LIMIT (Repeated)

The equation

___________________

means that whenever a hyperreal number x is infinitely close to but not equal to c,

f(x) is infinitely close to L.

ε, δ CONDITION FOR ________________

For every real number ε > 0 there is a real number δ > 0 which depends on ε such that whenever x is strictly within δ of c but not equal to c, f(x) is strictly within ε of L. In symbols, if 0< |x-c| < δ, then |f(x) - L|< ε.

In the ε, δ condition, the notion of being infinitely close to c is replaced by being strictly within δ of c, and being infinitely close to L is replaced by being strictly within ε of L. But why are there two numbers ε and δ, instead of just one? And why should δ depend on ε? Let us look at a simple example.

EXAMPLLE 1 Consider the limit _________________.

When x=0, the function f(x) = 1 + 10x²/x is undefined. When x is a real number close to

but not equal to 0, f(x) is close to 1.

Now let us be more explicit. How should we choose x to get f(x) strictly within ___ of 1?

To solve this problem we assume x is strictly within some distance δ of 0 and get

inequalities for f(x).

By the lemma, we must find a δ > 0 such that whenever

- δ < x < δ and x ≠ 0,

We have ______________________.

Assume - δ < x and x< δ.

Then -10δ < 10x and 10x<10δ

__________________________________ if x ≠ 0

__________________________________

1- 10δ < f(x) < 1+10δ.

If we set ________, then

_____________________________

This shows that

Whenever _______________________________

In other words,

_______________________________

A similar computation shows that for each ε > 0, if 0 < |x| < ε/10 then |f(x)-1| < ε. Thus

the ε, δ condition for _____________ (1+10x²/x)=1 is true, and, for a given ε, a

corresponding δ is δ = ε/10.

EXAMPLE 2 In the limit

________________________

find a δ > 0 such that whenever 0 < |x-2|< δ, |x² - 4| < ______.

By the Lemma, we must find δ > 0 such that whenever

2 - δ < x < 2+δ and x ≠ 2,

_________________.

Assume that 2 - δ < x and x < 2 + δ.

As long as 2 - δ and x are positive we may square both sides,

4 - 4δ +δ² < x² and x² < 4+4δ+δ²

4 + (-4δ + δ²) < x² and x² < 4+(4δ+δ²).

Now take δ small enough so that

___________________ and __________________.

For example, _____________ will do. Then

_____________________________.

Thus whenever 0 < |x-2| < _____, |x²-4| < _____.

Notice that any smaller value of δ, such as δ = ______, will also work.

In geometric terms, the ε, δ condition says that for every horizontal strip (of width 2ε) centered at L, there exists a vertical strip (of width 2δ) centered at c such that whenever x ≠ c is in the vertical strip, f(x) is in the horizontal strip. The graphs in Figure 5.8.2 indicate various horizontal strips and corresponding vertical strips. They should be examined closely.

There are also ε, δ conditions for one-sided limits and infinite limits. The three cases below are typical.

ε, δ CONDITION FOR ____________

For every real number ε > 0, there is a real number δ > 0 which depends on ε such that

whenever c < x < c+ δ, we have |f(x) - L| < ε.

Intuitively, when x is close to c but greater than c, f(x) is close to L.

ε, δ CONDITION FOR _____________

For every real number ε > 0, there is a real number B > 0 which depends on ε such

that whenever x > B, we have |f(x) - L| < ε.

Intuitively, when x is large, f(x) is close to L.

ε, δ CONDITION FOR _____________

For every real number A > 0, there is a real number B > 0 which depends on A such

that whenever x > B, we have f(x) < A.

Intuitively, when x is large, f(x) is close to L.

ε, δ CONDITION FOR _____________

For every real number A > 0, there is a real number B > 0 which depends on A such

that whenever x > B, we have f(x) > A.

Intuitively, when x is large, f(x) is large.

EXAMPLE 3 In the limit

____________________________,

find a real number B > 0 such that whenever t > B, (2 + 3/t) is strictly within 1/100 of 2.

To find B, we assume t > B and t > 0, and get inequalities for 2 + 3/t.

0 < t, t > B

_______, __________

_________, _____________

Now choose B so that 3/B ≤ 1/100. The number B = 300 will do.

It follows that whenever t > 300,

________________________,

and _____ is strictly within _______ of 2.

EXAMPLE 4 In the limit

____________________

find a B > 0 such that whenever x > B, x² - x > 10,000.

This time we assume x > B and get an inequality for x² - x. We may assume B > 1.

x > B > 1

x - 1 > B - 1 > 0

X(x-1) > B (B-1)

x² - x > B² - B.

Now take a B such that B² - B > 10,000. The number B=200 will do, because

(200)² - 200 = 39800. Thus whenever x > 200, x² - x > 10,000.

We conclude this section with the proof that the ε, δ condition is equivalent to the infinitesimal definition of a limit.

THEOREM 1

Let f be defined in some deleted neighborhood of c. Then the following are equivalent:

(i) _________________

(ii) The ε, δ condition for _____________ is true.

PROOF We first assume the ε, δ condition and prove that

_________________________

Let x be any hyperreal number which is infinitely close but not equal to c. To prove

that f(x) is infinitely close to L we must show that

for every real ε > 0, |f(x) - L| < ε.

Let ε be any positive real number, and let δ > 0 be the corresponding number in the ε, δ condition. Since x is infinitely close to c and δ > 0 is real, we have

0< |x-c| < δ.

By the ε, δ condition and the Transfer Principle,

|f(x) - L| < ε.

We conclude that f(x) is infinitely close to L. This proves that

_______________________.

For the other half of the proof we assume that

_______________________,

and prove the ε, δ condition. This will be done by an indirect proof. Assume that the ε, δ condition is false for some real number ε > 0. That means that for every real δ > 0 there is a real number x = x(δ) such that

(1) x ≠ c, |x-c| < δ, |f(x) - L | ≥ ε .

Now let δ1 > 0 be a positive infinitesimal. By the Transfer Principle, Equation (1) holds for δ1. Therefore x1 = x(δ1) is infinitely close but not equal to c. But since

|f(x1) - L | ≥ ε .

and ε is a positive real number, f(x1) is not infinitely close to L. This contradicts the equation

__________________________

We conclude that the ε, δ condition must be true after all.

The theorem is also true for the other types of limits.

The concept of continuity can be described in terms of limits, as we saw in Section 3.4. Therefore continuity can be defined in terms of the real number system only.

COROLLARY

The following are equivalent.

(i) f is continuous at c.

(ii) for every real ε > 0 there is a real δ > 0 depending on ε such that:

Whenever |x-c| < δ, |f(x) - f(c)| < ε.

PROOF Both (i) and (ii) are equivalent to

_________________________.

Intuitively, this corollary says that f is continuous at c if and only if f(x) is

close to f(c) whenever x is close to c.

PROBLEMS FOR SECTION 5.8

1 In the limit _________________ , find a δ > 0 such that whenever 0 < |x - 4 |< δ,

|10x-40| < 0.01.

2 In the limit __________________, find a δ > 0 such that whenever 0 < |x| < δ,

|(x²-4x) / 2x- (-2)| <0.1.

3 In the limit __________________, find a δ > 0 such that whenever 0 < | x-2 | < δ,

|1/x - 1/2| < 0.01.

4 In the limit __________________, find a δ > 0 such that whenever 0 < |x-(-3)| < δ,

| x³- (-27)| < 0.01.

5 In the limit __________________, find a δ > 0 such that whenever 0 < x < δ, ___.

6 In the limit __________________, find a δ > 0 such that whenever 2 < x < 2 + δ,

_____________________.

7 In the limit __________________, find a δ > 0 such that whenever 1-δ < x < 1,

_____________________.

8 In the limit __________________, find a δ > 0 such that whenever 2-δ < x < 2,

____________________ .

9 In the limit __________________, find a δ > 0 such that whenever 0 < |x| < δ,

x-² > 10,000.

10 In the limit __________________, find a δ > 0 such that whenever 0 < |x| < δ,

16/x4 > 10,000.

11 In the limit __________________, find a δ > 0 such that whenever 0 < t < δ,

10t > 100.

12 In the limit __________________, find a δ > 0 such that whenever 4 < t < 4+ δ,

1/(4-t) < -100.

13 In the limit __________________, find a δ > 0 such that whenever 0 < x < δ,

______ > 100.

14 In the limit __________________, find a δ > 0 such that whenever 0 < x < δ,

1/x³ >1000.

15 In the limit __________________, find a δ > 0 such that whenever 1-δ < x < 1,

1/(1-x²) >100.

16 In the limit __________________, find a δ > 0 such that whenever 2-δ < x < 2,

________________________.

17 In the limit __________________, find a B > 0 such that whenever t >B,

1/(1+4t) < 0.01.

18 In the limit __________________, find a B > 0 such that whenever t > B,

1/t² < 0.01.

19 In the limit __________________, find a B > 0 such that whenever t >B,

2t² - 5t > 1000.

20 In the limit __________________, find a B > 0 such that whenever t > B,

t³ + t² -5 > 1000.

21 In the limit __________________, find a δ > 0 such that whenever 1-δ < x < 1,

1/(1-x²) >100.

22 In the limit __________________, find a B > 0 such that whenever x < -B,

___________________.

□23 State the ε, δ condition for the limit ____________.

□24 State the ε, δ condition for the limit ____________.

□25 State the ε, δ condition for the limit ____________.

□26 Prove that _________ if and only if the ε, δ condition for this limit holds: For

every A > 0 there is a B > 0 such that whenever x > B, f(x) > A.