# Bernoulli's Rule

1. Apr 7, 2013

### Bachelier

In his treatment of L'hôpital/Bernoulli's rule (please see attached), Rudin before ineq. $(17)$ mentions that since the differentiable quotient

$\frac{f'(x)}{g'(x)} \rightarrow A$ as $x \rightarrow a$ and $A<r$ then there exists a pt $c \in (a,b) \ s.t. \ a<x<c \Rightarrow \ \frac{f'(x)}{g'(x)}<r$

Is it so because $x$ approaches $a$ that's why he used $a<x<c$ instead of $c<x<b$

and why this $c$ in the first place? What's wrong with just saying, $\exists x \in (a,b)$ etc

#### Attached Files:

• ###### Rudin thm 5.13.png
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Last edited: Apr 7, 2013
2. Apr 8, 2013

### mathman

The theorem is not there exists an x in the interval, but rather for all x, a < x <c, f'/g' < r.

3. Apr 21, 2013

True. Thanks