L'Hospitals Rule

1. Aug 7, 2005

iNCREDiBLE

I have an examina soon and I need help with following proof. I don't know TEX that good so I'm attaching a screenshot from word instead.

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2. Aug 7, 2005

Suppose that f and g are contiunuous on [a,b], differentiable on (a,b), that $$c{\in}[a,b]$$, and that $$g(x){\not}=0$$ for $$x{\in}[a,b]$$, $$x{\not}=c$$.
Let $$A:=\lim_{x{\to}c}f$$ and $$B:=\lim_{x{\to}c}g$$.
In adition to the suppositions, let g(x)>0 for $$x{\in}[a,b]$$, $$x{\not}=c$$.
(a)If A>0 and B=0, prove that we must have $$\lim_{x{\to}c}\frac{f(x)}{g(x)}=\infty$$
(b)Also, if A<0 and B=0, prove that we must have $$\lim_{x{\to}c}\frac{f(x)}{g(x)}=-\infty$$