第5.2节 罗必达法则

5.2 L′ HOSPITAL′S RULE

Suppose f and g are two real functions which are defined in an open interval containing a real number a, and we wish to compute the limit

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Sometimes the answer is easy. Assume that the limits of f(x) and g(x) exist as x→a,

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If M≠0, then the limit of the quotient is simply the quotient of the limits,

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This is because for any infinitesimal Δx ≠ 0,

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If L ≠ 0 and M = 0, then the limit

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does not exist, because when Δx ≠ 0 is infinitesimal, f(a+Δx) has standard part L ≠ 0 and g(a+Δx) has standard part 0.

But what happens if both L and M are 0? In some cases a simple algebraic manipulation will enable us to compute the imit. For example,

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even though both the numerator x² -1 and the denominator x+1 approach 0 as x approaches -1.

In other cases I’ Hospital’s Rule is useful in computing limits of quotients where both L and M are 0. Before stating I’ Hospital’s Rule, we introduce the notion of a neighborhood of a point c (Figure 5.2.1).

Figure 5.2.1

DEFINITION

By a neighborhood of a real number c we mean an interval which contains c as an interior point.

The set formed by removing the point c from a neighborhood I of c is called a deleted neighborhood of c. Thus a deleted neighborhood is the set of all points x in I such that x ≠ c.

L’ HOSPITAL’S RULE FOR 0/0

Suppose that in some deleted neighborhood of a real number c, f ′ (x) and g ′(x) exist and

g ′(x) ≠0. Assume that

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If ____________ exists or is infinite, then

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(See Figure 5.2.2.) Usually the limit will be given by

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and in this case the proof is very simple.

Figure 5.2.2 L’ Hospital’ s Rule

PROOF IN THE CASE ______________________

Let Δx be a nonzero infinitesimal. Then f(c) = 0, g(c) = 0, and

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Taking standard parts we get

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Intuitively, for x ≈ c the graphs of f(x) and g(x) are almost straight lines of slopes f ′(c), g′(c) passing through zero, so the graph of f(x)/ g(x) is almost the horizontal line through

f ′(c) / g′(c) (Figure 5.2.3).

The equation

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is not always true. For example, g′(c) might be zero or undefined.

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is sometimes another limit of type 0/0, that is,

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When this happens, 1’ Hospital’s Rule can often be reapplied to ______________f ′(x)/g′(x).

The proof of 1’ Hospital’s Rule in general is fairly long and uses the Mean Value Theorem. It will not be given here.

Here are some examples showing how the rule can be applied.

EXAMPLE 1 Find _____________________

Both (1/x) -1 and ______ approach 0 as x approaches 1. The limit is thus of the form 0/0. Using 1’ Hospital’s Rule,

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EXAMPLE 2 Find _____________________

The limit is of the form 0/0. The limit of f ′(x) / g′(x) as x →0 is ∞.

_______________________________

Thus by 1’ Hospital’s Rule,

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EXAMPLE 3 Find ____________________

This limit is not in a form where we can apply 1’ Hospital’s Rule. We must first use algebra to put it in another form,

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By elementary computations, __________________________

Using 1’ Hospital’s Rule,

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We then add the limits to get the desired answer,

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EXAMPLE 4 Find ______________________

This limit is of the form 0/0. When 1’ Hospital’s Rule is used the limit is still of the form 0/0. But when it is used a second time we can compute the limit.

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L’ Hospital’s Rule also holds true for other types of limits. That is, it holds true if x→c is everywhere replaced by one of the following.

x→c+, x→c -, x→ ∞, x→ - ∞.

EXAMPLE 5 Find ___________

The limit as x → 0 does not exist because ____ is defined only for x > 0. However, the one-sided limit as x → 0+ has the form 0/0 and can be found by 1’ Hospital’s Rule.

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A second form of 1’Hospital’s Rule deals with the case where both f(x) and g(x) approach ∞ as x approaches c.

L’ HOSPITAL’S RULE FOR ∞/∞

Suppose c is a real number, and in some deleted neighborhood of c, f ′(x) and g′(x) exist and g′(x) ≠ 0. Assume that

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If _________________exists or is infinite, then

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The rule for ∞/ ∞ is exactly the same, word for word, as the rule for 0/0, except that 0 is replaced by ∞. We omit the proof, which is more difficult in the case ∞/ ∞. Actually, the assumption

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is not needed.

Again, 1’ Hospital’s Rule for ∞/ ∞ also holds for the other types of limits,

x→c+, x→c -, x→ ∞, x→ - ∞.

EXAMPLE 6 Find __________________.

By 1’ Hospital’s Rule for ∞/∞,

_______________________________

Warning: Before using 1’Hospital’s Rule , check to see whether the limit is of the form 0/0 or ∞/∞. A common mistake is to use the rule when the limit is not of one of these forms.

EXAMPLE 7 Find ____________

The limit has the form 0/1, so 1’ Hospital’s Rule does not apply.

Correct: ______________________

Incorrect:

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PROBLEMS FOR SECTION 5.2

In Problems 1-34, evaluate the limit using 1’Hospital’s Rule.

1

3

5

7

9

11

13

15

17

19

21

23

25

27

29

31

33

In Problems 35-52, evaluate the limit by 1’Hospital’s Rule or otherwise.

35

37

41

45

47

49

51

□53 Suppose f and g are continuous in a neighborhood of a and g(a) ≠ 0. Show that

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