How to use the $(\epsilon-\delta)$-definition to prove the existence or non-existence of a limit?

Question

For example, consider this question-

Use the $(\epsilon, δ)$-definition to prove the existence or non-existence of the following limit- $$f : R → R, f(x) := [x]$$

$$ \lim_{x→0} f(x)$$

Here we do not know apriori if the limit exists or not. Now I am confused about whether I should start with the assumption that the limit exists or instead assume it does not exist (I mean which is an easier way to show it). Also, should I try to assume the opposite of the correct statement to be true and try to use a counterexample or should I try to show the correct statement it in a direct way?

Answer 1

You have to assume that the limit exists and get a contradiction. Suppose $[x] \to l$ as $ x\to 0$. Then there exits $\delta >0$ such that $|[x]-l| <\frac 1 2$ whenever $|x|<\delta$. Since we may replace $\delta$ by any smaller number we may suppose $\delta <1$. Take $x=\delta /2$ to see that $|0-l|<\frac 1 2$. Then take $x=-\delta /2$ to get $|(-1)-l| <\frac 1 2$. Combine these two to get $1<1$, a contradiction.

Answer 2

A simpler way is to check by $(\epsilon, δ)$ definition that

• $\lim_{x→0^+} f(x)=0$
• $\lim_{x→0^-} f(x)=-1$