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