If $f(x)>0$ near $x_0$ and $\lim_{x\to x_0}f(x)$ exists, then why is it always $\lim_{x\to x_0}f(x)\geq0$? 
If $f(x)>0$ near $x_0$ and $\lim_{x\to x_0}f(x)$ exists, then why is it always $\lim_{x\to x_0}f(x)\geq0$?

The basic limit theorems, state that If $\lim_{x \to x_0}f(x)>0$, then $f(x)>0$, near $x_0$. Why does the opposite of that have to contain an equality?

Consider the function $f:\mathbb{R}\to\mathbb{R}$ given by $f(x)=|x|.$ Then $f(x)>0$ for all $x \neq 0.$ But $\lim_{x\to 0} f(x)=0.$
However, if $\lim_{x\to x_0}=L>0,$ then corresponding to $\frac{L}2>0,$ there exists $\delta >0$ such that $|f(x)-L|<\frac{L}{2}$ whenever $|x-x_0| <\delta.$ So near $x_0,$ we have $f(x)>L-\frac{L}2=\frac{L}{2}>0.$
