ppprove or desprove question

prove or display a desproving example:
A)
there is f and g which are defined $\displaystyle x_{0}-\delta<x<=x_{0}+\delta$ for which $\displaystyle lim_{x->x_{0}}f(x)g(x)=\infty$.
if 0<g(x)<1 for $\displaystyle x\in N_{\delta}^{*}(x_{0})$ then $\displaystyle lim_{x->x_{0}}f(x)=\infty$
B)
if f monotonickly decreasing in $\displaystyle [0,\infty)$ and $\displaystyle lim_{x->\infty}f(x)=0$ then f(x)>0 for all $\displaystyle x\in[0,\infty)$
C)if f differentiable in (a,b) so $\displaystyle lim_{x->a^{+}}f(x)=lim_{x->b^{-}}f(x)=1$ then there is $\displaystyle c\in(a,b)$ so $\displaystyle f'(c)=0$
D)if f is integrabile in [a,b] then there is x in [a,b] so $\displaystyle \int_{a}^{x}f(t)dt=\int_{x}^{b}f(t)dt$

regarding a:
i dont know from the start how to know intuetivly whether its true or false
?