On Some Types of Limit

In this paper, we consider the concept of limit, one of the basic concepts of mathematical analysis. At a point $a\in{\mathbb{R}}$, the limit of a function $f$ from $A\subset\mathbb{R}$ to $\mathbb{R}$ is $L\in{\mathbb{R}}$ if and only if there exists $\delta>0$ such that the set $$\left\{x\in\left(\left(a-\delta,a+\delta\right)\backslash\left\{a\right\}\right)\cap A:\lvert f\left(x\right)-L\rvert \ge\varepsilon\right\}$$ is the empty set for each $\varepsilon>0$. This study investigates what happens when the above set is not empty. More clearly, when we take the above set as finite or a set whose accumulation points is empty, we prove that the limit concept does not change. However, when we take the above set as countable or a set of measure zero, we prove that they are new limit concepts. Also, we show that these new limit concepts have the properties of existence, uniqueness, additivity, multiplicativity, etc. Finally, by giving some examples, we compare the limit concepts we have obtained with the previous ones.