Why is the limit of a function $f(x)$ defined as $$f(x)-L<\varepsilon$$$$x-a<\delta$$but not like $$f(x)-L \leq \varepsilon$$ $$x-a \leq \delta$$What problems would arise if it was defined like this as compared to the standard definition?
P.S.- Some may point out these as silly questions, but I want to understand every bit that I read, not just know. Thanks.
Assume that $A \subset \mathbb R$ and $f: A \to \mathbb R$ and $a$ is a limit point of $A$. Let $L \in \mathbb R$.
You can show that $\lim_{x \to a} f(x) = L$ if and only if for all $\epsilon > 0$ there exists $\delta > 0$ such that if $x \in A$ and $0 < | x - a | \leq \delta$ then $| f(x) - L | \leq \epsilon$.
So, the alternate definition you're suggesting is equivalent to the usual definition.
