Limit True/False True or False?
If $f(x) < g(x)$ on some deleted neighborhood of point $c$, and if $\lim_{x \rightarrow c} f(x)$ and $\lim_{x \rightarrow c} g(x)$ exist, then $\lim_{x \rightarrow c} f(x) < \lim_{x \rightarrow c} g(x)$.
I've been given a theorem where I believe this holds for all "$\leq$" as opposed to "$<$". From this, I tend to believe its true, but how would I prove that?

False.
For example, $x^2<2x^2$ whenever $x\ne0$, but their limits as $x\to0$ are equal.
