General Limit Question

Question

I'm taking an online Calc 1 class to review concepts, as my math skills are a little rusty. This was one of the questions, $$ f(x) > a, \text{ for } a \in \mathbb{Z} \text{, } \forall x \in \mathbb{R}\text{, and } \lim_{x\to 0} f(x) \text{ exists. So} \lim_{x\to 0} f(x) > a. \text{ True or False.} $$ The answer is false, in general. But why? It must be true if $f(x)$ is continuous, so it must have something to do with a discontinuous $f(x)$.

Answer 1

The answer is false in general not only for the fact that $f$ can be discontinuous, but also for the fact that the limit operation does not preserve strict inequality. For example, choosing $a\in\mathbb{N}, a\neq0$ we have that $$ f(x)=a\left(1+e^{-\frac{1}{|x|}}\right)>a\quad \forall x\in\mathbb{R}\setminus 0 $$ but $$ \lim_{x\to0} f(x)=a $$

Answer 2

Let

$$ f(x)=\left\{\begin{align} 3&\quad x\in \mathbb{Z}\\0& \quad \text{else}\end{align} \right\} $$

Then if $a=2$, we have $f(x)>a$ for $x\in \mathbb{Z}$, but $\lim_{x\rightarrow 0}f(x)=0$.