# L'Hospitals Rule

by iNCREDiBLE
Tags: lhospitals, rule
 P: 266 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 $$\lim_{x{\to}c}\frac{f(x)}{g(x)}=-\infty$$