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Bachelier
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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
##\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
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