8.6 L'Hôpital's Rule
| 8.6 |
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| 1. The Cauchy Mean-Value Theorem |
For a review of the mean-value theorem see Section 5.1 Theorem 5.1.
where D2, C2, G2, and F2 are the counterparts for point c of D, C, G, and F respectively.
The property expressed by Eq. [1.1] is called the Cauchy mean-value theorem or the generalized mean-value theorem,
because from Eq. [1.1] we get:
Also note that Eq. [1.1] says that the ratio of the total changes of f and g on [a, b] is equal to the ratio of their
instantaneous rates of change at c.
Theorem 1.1 – The Cauchy (Or Generalized) Mean-Value Theorem
Proof
Let:
h(x) = ( f(b) – f(a))g(x) – (g(b) – g(a)) f(x).
Clearly h is continuous on [a, b] and differentiable on (a, b). Also:
h(a) = ( f(b) – f(a))g(a) – (g(b) – g(a)) f(a) = f(b)g(a) – f(a)g(b)
and:
h(b) = ( f(b) – f(a))g(b) – (g(b) – g(a)) f(b) = f(b)g(a) – f(a)g(b) = h(a).
So, by Rolle’s theorem, there exists c in (a, b) such that h‘(c) = 0. We have:
h‘(x) = ( f(b) – f(a))g‘(x) – (g(b) – g(a)) f ‘(x),
yielding:
h‘(c) = ( f(b) – f(a))g‘(c) – (g(b) – g(a)) f ‘(c).
Thus:
( f(b) – f(a))g‘(c) = (g(b) – g(a)) f ‘(c).
EOP
The Cauchy mean-value theorem is also called the generalized mean-value theorem.
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In Section 1.1.3 we encountered limits of the indeterminate quotient form of type 0/0, and in Section 1.1.7 we
Remark that the right-hand side is the quotient f ‘/g‘ of the derivatives f ‘ and g‘, not the derivative of the quotient f/g,
which is ( f ‘g – fg‘ )/g2.
| If: |
Proof
The proofs for one-sided limits are contained in this proof. The proofs for limits at infinity are similar to this proof.
EOP
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Example 3.1
Find:
Solution
EOS
Applying More Than Once
Example 3.2
Find:
Solution
EOS
Arranging Into Forms For The Rule
Example 3.3
Evaluate:
where a is a positive constant.
Solution
EOS
Trying To Simplify Before Trying To Apply The Rule
In Example 3.3, the limit:
So before trying to apply the rule we should first try to simplify the limit’s expression as much as possible. After simplification it may or may not turn out that there’s no more indeterminate form. If you just keep applying the rule
without trying to simplify first, the process may never end.
For Functions Of The Form ( f (x)) g(x)
Example 3.4
Solution
EOS
When Not To Use The Rule
Trivially:
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| In Section 7.2 Part 10 |
For any given integer n > 0 we have:
EOP
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Solution
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Solution
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Solution
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4. Consider this “ famous” trigonometric limit:
b. Can the computation in part a be considered as another proof of that limit? Why or why not?
Solution
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5. What’s wrong with the following “ proof ” of the Cauchy mean-value theorem?
“ Since f is continuous on [a, b] and differentiable on (a, b), the mean-value theorem implies that there exists c in
(a, b) such that:
Solution
The “ proof ” assumes that there exists the same c that satisfies both equations [A] and [B], which is incorrect.
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