8.6 
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1. The Cauchy MeanValue Theorem 
For a review of the meanvalue theorem see Section 5.1 Theorem 5.1.
where D_{2}, C_{2}, G_{2}, and F_{2} are the counterparts for point c of D, C, G, and F respectively.
Fig. 1.1

The property expressed by Eq. [1.1] is called the Cauchy
meanvalue theorem or the generalized meanvalue
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) MeanValue 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 meanvalue theorem is also called the generalized meanvalue 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 righthand 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' )/g^{2}.
If:

Proof
The proofs for onesided limits are contained in this proof.
The proofs for limits at infinity are similar to this proof.
EOP
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Find:
EOS
Find:
Evaluate:
where a is a positive constant.
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}^{)}
Trivially:
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For any given integer n > 0 we have:
EOP
Problems & Solutions 
Solution
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
5. What's wrong with the following proof of the Cauchy meanvalue theorem?
Since f
is continuous on [a, b] and differentiable
on (a, b), the meanvalue
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.