If $f(x)I have this question:

Let $f(x)→A$ and $g(x)→B$ as $x→x_0$. Prove that if $f(x) < g(x)$ for all $x∈(x_0−η, x_0+η)$ (for some $η > 0$) then $A\leq B$. In this case is it always true that $A < B$?

I've tried playing around with the definition for limits but I'm not getting anywhere. Can someone give me a hint on where to start?

To show it is not always the case that $A<B$, you can come up with an example to show it is possible for $A = B$. So if we let $f(x) = (\frac{1}{x})^2$ and $g(x) = (\frac{1}{x})^4$, we know $f(x) < g(x)$ $\forall x \in (-1,1)$. And we know $\lim_{x \to 0} f(x) = \lim_{x \to 0} g(x) = \infty$ so it is possible for $A=B$. 
Note the interval I gave you above is of the form $(x_0-η,x_0+η)$, which is centered about $x_0$ (distinguishing this from other answers).
