What happens if $\epsilon$ and $\delta$ are zero in the definition of the limit?

Question

Can someone explain what happens when epsilon and delta are equal to zero in the definition of limits using epsilon-delta?

Answer 1

Reminder of the definition: $\lim_{x \to a}f(x) = L$ means: for all $\epsilon >0$, there exists a $\delta >0$ such that $$0<|x - a| < \delta \Rightarrow |f(x) - L| < \epsilon $$

If $\delta > 0$ and $\epsilon = 0$ it means that the value of the function should be exactly the target value in the neighbourhood of the point where you take the limit.

If $\delta = 0$, then the "neighbourhood" is just one point, and the definition of the limit reduces to "if $x=y$, then $f(x)=f(y)$"; this is useless.